Neural Representations of Pitch:
Role of Peripheral Frequency Selectivity
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MASSACHUSETTS INSTITUTE
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by
Leonardo Cedolin
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Laurea in Electrical Engineering,
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DOCTOR OF PHILOSOPHY
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HEALTH SCIENCES AND TECHNOLOGY
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
FEBRUARY 2006
©2006 Massachusetts Institute of Technology
All rights reserved.
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Division of Health Sciences and Technology
January 27, 2006
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Neural Representations of Pitch:
Role of Peripheral Frequency Selectivity
By
Leonardo Cedolin
Laurea in Electrical Engineering, Politecnico di Milano, 1999
Submitted to the Division of Health Sciences and Technology on January 27, 2006
in partial fulfillment of the requirements for the degree Doctor of Philosophy in
Health Sciences and Technology
Abstract
Investigating the neural mechanisms underlying the perception of the pitch of harmonic
complex tones is of great importance for many reasons. Changes in pitch convey melody in
music, and the superposition of different pitches is the basis for harmony. Pitch has an
important role in speech, where it carries prosodic features and information about speaker
identity. Pitch plays a major role in auditory scene analysis: differences in pitch are a major
cue for sound source segregation, while frequency components that share a common
fundamental frequency (F0) tend to be grouped into a single auditory object.
In psychophysics, a positive correlation is commonly observed between the estimated
"resolvability" of individual harmonics of complex tones, assumed to depend primarily on
the frequency selectivity of the cochlea, and the strength of the corresponding pitch percepts.
In this study, possible
neural codes for the pitch
of harmonic
complex tones were
investigated in the auditory nerve of anesthetized cats, with particular focus on their
dependence on cochlear frequency selectivity, which was measured directly using both
complex tones and band-reject noise.
A "rate-place" representation of pitch, based on cues to peripherally-resolved harmonics
in profiles of average discharge rate along tonotopically-arranged neurons, was compared to
a "temporal" representation, based on periodicity cues in the distributions of interspike
intervals of the entire auditory nerve. Although both representations were viable in the range
of F0s of cat vocalizations, neither was entirely satisfactory in accounting for psychophysical
data. The rate-place representation degraded rapidly with increasing stimulus level and could
not account for the upper limit of the perception of the pitch of missing-F0 in humans, while
the interspike-interval representation could not predict the correlation between
psychophysical pitch salience and peripheral harmonic resolvability.
Therefore, we tested an alternative, "spatio-temporal" representation of pitch, where cues
to the resolved harmonics arise from the spatial pattern in the phase of the basilar membrane
motion.
The spatio-temporal
representation
was relatively
stable with level and was
consistent with an upper limit for the pitch of missing-F0, thus becoming the strongest
candidate to explain several major human pitch perception phenomena.
Thesis Supervisor: Bertrand Delgutte
Title: Associate Professor of Otology and Laryngology and Health Sciences and Technology
Harvard Medical School
2
Acknowledgements
Questo lavoro
dedicato ai miei genitori, Carla and Luigi Cedolin.
Grazie, mammae papa, per 'affetto, il sostegno morale e l'incitamento che mai mi avete
fatto mancare nonostante la lontananza, e per essere sempre stati il mio modello di
onesta, impegno e perseveranza.
I am deeply and truly grateful to my advisor Bertrand, for constantly steering me back
onto the right path, and to Nelson, for giving me this life-changing opportunity. Thanks!
Among many others, the following people contributed to making my academic life and
my adventure in Boston enjoyable and unforgettable. Thanks to all of you !
Andrew Oxenham, Chris Shera and Garrett Stanley, members of my thesis committee;
Julie Greenberg and John Wyatt; the entire Speechand Hearing faculty; the HST
staff at E-25; Peter Cariani; Sergio Cerutti e Emanuele Biondi; Dianna, Laura, Mike R.,
Connie, Leslie, Stephane, Kelly, Ish, Frank, Chris and Wen at EPL; Dr. Merchant, Dr.
Halpin, C)r. Varvares and Dr. Nadol; Jocelyn, Erik, Suzuki and Waty ("coffee? Yeah,
why not?"); my awesome lunch buddies Zach, Brad and Dave; Courtney, Chandran, and
Anna ("is my music too loud? is the light too dim?"); Craig ("The strongest man alive"),
Sasha, Ken ("I'll put anything on pizza"), Teresa; Keith and Darren ("don't talk about
Fight Club"); Josh B. ("amici!!!!")and Jill; Hector, Isabel and Joey; the softball tribe
and Amy at Shay's; Ashish and Minnan; the Mays, the Rizzos and the Stasons; Ozzie,
Barclay and Finnigan; Jamie, Kate, Mark, Meri, Cis, George and Kari; Chris Nau, Greg
and Ray; Chris B. ("I'll gamble"); "Orange" Jeroen, "Crazy" Hannah, Eric P., Juan-ita,
Michi, Jeff "Gold Man"; my great HMS friends Kedar, Theo, Irina, Mindy and Sarah;
Jean; my roommates (and their
families):
"hawaii" Chris E., "kimchi is gone" Kwang-
Wook, "VV" Virgilio, Mia (and Cathy and the Irish gang); my amazing Italian buddies Ila,
Ponchia, Zed, Claudia, Sbira, brago e Pellegra; the 67a Dana gang: Paul, Remo
("whazzuuup") and Steak ("hey-hey"); Bruce ("nice hat dude"); my brothers Martino,
Sridhar, Mike C. and Noop; Ali, my funny, smart, sweet, loving Bambina.
3
Table of Contents:
Abstract ................................................................................................
Acknowledgements
..................................................................................
Introduction .............................................................................................
2
3
5
1. Pitch of Complex Tones: Rate-Place and Interspike-Interval Representations in the
Auditory Nerve ..................................................................................
11
Figures, Chapter 1 ..............................................................................
41
2. Spatio-Temporal Representation of the Pitch of Complex Tones in the Auditory Nerve 53
Figures, Chapter 2 ..............................................................................
3. Frequency Selectivity of Auditory-Nerve Fibers Studied with Band-Reject Noise
Figures, Chapter 3 ..............................................................................
4. Summary and conclusions
.....................................................................
5. Implications and future directions
...........................................................
References .............................................................................................
4
75
.... 89
104
119
124
129
INTRODUCTION
Harmonic complex tones are the sum of single sinusoidal sounds ("pure tones") whose
frequencies are all integer multiples of a common fundamental frequency (F0). Harmonic
complex tones are an extraordinarily important class of sounds found in natural
environments. For example, harmonic complex tones are produced by the vibrations of the
vocal folds, source of "voiced" sounds in human speech. Sounds produced by a wide variety
of musical instruments are also harmonic, as often are animal vocalizations.
Typically, individual frequency components of a harmonic complex tone are not
perceived by human listeners as separate entities, but they are "grouped" together into a
single auditory percept, commonly known as pitch. Pitch is defined as "that attribute of
auditory sensation in terms of which sounds may be ordered on a scale extending from low to
high" (ANSI 1973). Many classes of sounds can evoke a pitch sensation, including for
example pure tones, complex tones and even some natural or appropriately manipulated
types of noise. The pitch evoked by a harmonic complex tone with a fundamental frequency
F0 typically coincides with the pitch evoked by a pure tone with frequency equal to F0. For
FOs up to 1400 Hz (Moore 1973), this holds even when the component at the F0 is not
physically present in the sound (Schouten 1940) or is masked with noise (Licklider 1954).
This phenomenon is known as "pitch of the missing fundamental". Interestingly, a pitch at
the F0 can still be heard even when a harmonic complex lacks a number of components, and
in this case the strength of the perceived pitch depends strongly on which harmonics are
missing (e.g. Houtsma and Smurzynski 1990).
Pitch plays many important roles in speech, where it provides the foundations for prosody
and information used to segregating simultaneous speakers (Darwin and Hukin 2000).
tone languages such as Mandarin Chinese, pitch also carries lexical information.
In
Changes in
pitch convey melody in music, and the superposition of different pitches is the basis for
harmony. Pitch plays a major role in auditory scene analysis: differences in pitch are a major
cue for sound source segregation, while frequency components that share a common
fundamental tend to be grouped into a single auditory object (Bregman 1990). Pitch may
also be of great importance for animals to process conspecific vocalizations, often consisting
of harmonic complex tones. Moreover, the pitch of the missing fundamental is known to be
5
perceived not only by humans, but also by cats (Heffner and Whitfield 1976), monkeys
(Tomlinson and Schwartz 1988) and birds (Cynx and Shapiro 1986). These observations
support the use of animal models (in our case the cat) to study neural representations of pitch.
The main goal of this thesis was to study possible neural mechanisms underlying the
perception of the pitch of harmonic complex tones, with particular focus on seeking a neural
representation of pitch that is consistent with major trends observed in human psychophysics
experiments, as no current physiologically-based
model can fully account for the whole body
of psychophysical data available on pitch perception.
Despite having been investigated for over a century (Seebeck 1841; Ohm 1843), the exact
nature of the neural coding of pitch is still debated. The peripheral auditory system can, in
principle, generate two types of cues to the pitch of harmonic complex tones. On one hand,
the harmonicity of the frequency spectrum (at F0) can be encoded thanks to the tonotopic
mapping and frequency analysis performed by the cochlea.
On the other hand, harmonic
sounds with the same fundamental frequency F0 also share the same waveform periodicity
(at TO =
/F0), reflected in regularities in the precise timings of neural action potentials
thanks to phase locking.
The peripheral
auditory system can be thought of as a bank of bandpass filters
(commonly referred to as "auditory filters"), which represent the mechanical frequency
analysis performed by the cochlea. When the frequency separation between two harmonics
exceeds the bandwidth of the auditory filters, these harmonics result in local maxima (at
locations tuned to the harmonics) and minima (at locations tuned to frequencies in between
those of the harmonics) in the spatial pattern of basilar membrane motion, reflected in the
resulting pattern of activation of the auditory nerve (AN). In this case, the harmonics are said
to be "resolved" by the auditory periphery.
On the other hand, when two or more harmonics
fall within the pass-band of a single peripheral filter, they are said to be "unresolved".
The
bandwidths of the auditory filters are known to increase with their center frequency (e.g.
Kiang et al. 1965; Shera et al. 2002), so that only low-order harmonics are resolved.
"Spectral" or "place" models of pitch perception propose that the pitch of complex tones
containing resolved harmonics may be extracted by matching the pattern of activity across a
tonotopic neural map to internally stored harmonic templates (Cohen et al. 1994; Goldstein
1973; Terhardt 1974; Wightman 1973).
6
However, also harmonic complex tones consisting entirely of high-order, unresolved
harmonics evoke a pitch percept. Unresolved harmonics do not produce local maxima in the
spatial pattern of activation along the cochlea, thereby excluding a purely spectral, templatebased, explanation for pitch extraction.
On the other hand, unresolved harmonics can
produce temporal cues to pitch because any combination of unresolved harmonics has a
period equal to that of the complex tone (TO). These periodicities, reflected in neural phase
locking, result in interspike intervals of individual AN fibers most frequently corresponding
to TO and its integer multiples.
"Temporal"
models of pitch perception propose that
periodicity cues to pitch may be extracted by an autocorrelation-type
1951; Meddis
and Hewitt
1991; Moore
equivalent to an all-order interspike-interval
1990; Yost
mechanism (Licklider
1996), which is mathematically
distribution for neural spike trains. This model
can in principle account also for the perception of the pitch of only resolved harmonics: since
TO is an integer multiple of the period of any of the harmonics, by "pooling" (summing)
autocorrelation functions (or, equivalently, all-order interspike-interval distributions) across
frequency channels one can extract the information about the common periodicity (Meddis
and Hewitt 1991; Moore 1990).
It cannot be ruled out that the true neural code for pitch might be neither purely placebased nor purely temporal. On one hand, the co-existence of two separate representations (a
place representation for resolved harmonics and a temporal representation for unresolved
harmonics) has been hypothesized (Carlyon and Shackleton 1994). On the other, more than
one plausible "spatio-temporal"
mechanism have been proposed, by which a combination of
place and temporal information might be used to identify individual components of a
harmonic complex tone.
For example, Srulovicz and Goldstein (1983) hypothesized that
individual harmonic frequencies may be extracted by passing the interspike-interval
distribution of each AN fiber through a filter matched to the period of the characteritisc
frequency (CF) of that fiber. Others (Loeb et al. 1983; Shamma 1985a) have suggested that
cues to spectral features of a sound could emerge when comparing the timings of spikes in
AN fibers with neighboring CFs, due to changes in the velocity of propagation of the
cochlear traveling wave.
In particular, the idea for an alternative, spatio-temporal representation of pitch stems
from the observation that the fastest variation with cochlear place of the phase of basilar
7
membrane motion in response to a pure tone occurs at the cochlear location tuned to the tone
frequency (Anderson 1971; Pfeiffer and Kim 1975; van der Heijden and Joris 2003).
At
frequencies below the limit of phase-locking, this rapid spatial change in phase of the basilar
membrane motion results in a rapid variation of the latency of the response of auditory-nerve
fibers with characteristic frequency, thus generating a cue to the frequency of the pure tone
(Shamma 1985a). Our hypothesis is that, for a harmonic complex tone, rapid changes in
phase may occur at each of the spatial locations tuned to a resolved harmonic.
Hence,
latency cues to the frequencies of each of the resolved harmonics of a complex tone would be
generated by the wave of excitation produced in the cochlea by each cycle of the stimulus, as
the wave travels from the base to the apex.
In principle, these spatio-temporal cues to
resolved harmonics can be extracted by a central neural mechanism sensitive to the relative
timing of spikes in AN fibers innervating neighboring cochlear locations, effectively
converting spatio-temporal cues into rate-place cues, which in turn can be combined to
generate a pitch percept by a template-matching mechanism.
The concept of harmonic resolvability has been introduced as a possible explanation for a
variety of phenomena observed in human studies of the perception of pitch. For example,
human subjects with normal hearing perform much better in detecting small differences in
the F0 of harmonic complexes in the presence of low-order harmonics than in their absence
(Bernstein and Oxenham 2003b; Houtsma and Smurzynski
1990).
This difference, in
combination with the increase in bandwidth of auditory filters with their center frequency, is
usually attributed to a neural mechanism more sensitive to resolved harmonics than to
unresolved harmonics. Hearing impaired listeners with sensorineural hearing loss exhibit a
decreased performance
in detecting small F-differences
(e.g. Moore and Peters
1992;
Bernstein and Oxenham 2006), at least partly as a result of their impaired cochlear frequency
selectivity (Bernstein and Oxenham 2006), thought in turn to result in a decreased degree of
harmonic resolvability. Moreover, the pitch evoked by exclusively high-order harmonics
depends strongly on the phase relationships among the partials, while pitch evoked by loworder harmonics is phase-invariant (Carlyon and Shackleton 1994; Houtsma and Smurzynski
1990). Since a dependence on phase is expected only for harmonics interacting within single
cochlear filters, these results can also be explained in terms of harmonic resolvability.
8
Human psychophysics experiments present the limitation that the sharpness of tuning of
the mechanical frequency analysis performed by the auditory periphery, assumed to
determine whether harmonics are resolved or not, cannot be estimated directly. Specifically,
the extent to which psychophysical measures of peripheral tuning sharpness are influenced
by more central stages of neural processing is unknown.
In this thesis, the strengths and
limitations of each of three possible neural representations of pitch mentioned above were
investigated in the auditory nerve of anesthetized cats and discussed principally in terms of
their dependence on harmonic resolvability in the species studied, which was directly
quantified for single auditory-nerve fibers in Chapter 1.
Chapter 1 explores, at the level of the cat auditory nerve, the viability and effectiveness
of a rate-place representation of the pitch of missing-fundamental harmonic complex tones,
based on the harmonicity of their frequency spectrum, the tonotopic structure of the cochlea
and harmonic resolvability. The results are compared to those obtained for a temporal
representation of pitch, based on waveform periodicity and on neural phase-locking reflected
in pooled all-order interspike-interval distributions.
The fundamental frequencies used
varied over a wide range (110 - 3520 Hz), extending over the entire F-range
of cat
vocalizations [500 - 1000 Hz (Brown et al. 1978; Nicastro and Owren 2003; Shipley et al.
1991)] and beyond. The results of this study indicate that, although both rate-place and
interspike-interval representations of pitch are viable in the cat AN over the entire F0-range
of conspecific vocalizations, neither is entirely consistent with human psychophysical data.
The rate-place representation predicts the greater strength and the phase-invariance of the
pitch of resolved harmonics, but it degrades rapidly with stimulus level and it fails to account
for the existence of an upper limit to the perception of the pitch of the missing fundamental
in humans. On the other hand, a main limitation of the interspike-interval representation lies
in the fact that it predicts greater strength for unresolved harmonics than for resolved
harmonics, in sharp contrast with human psychophysics results.
Chapter 2 presents for the first time a physiological test of a spatio-temporal
representation of the pitch of harmonic complex tones, based on the availability of temporal
cues to resolved harmonics
at the specific cochlear locations
tuned to the harmonic
frequencies. Effectiveness and strength of the spatio-temporal representation were first
studied in single AN fibers, and then these results were re-interpreted
9
as a function of
fundamental frequency using the scaling invariance principle (Zweig, 1976) in cochlear
mechanics.
The results of this study indicate that the spatio-temporal representation
overcomes the limitations of both a strictly spectral and of a strictly temporal representation
in accounting for trends observed in many studies of human psychophysics.
Chapter 3 tests the physiological basis of a method widely used in psychophysics to
estimate peripheral frequency selectivity: the notched-noise method (Patterson 1976). Like
other similar methods [for example psychophysical tuning curves, using pure tones (Moore
1978), or methods using rippled noise (Houtgast 1972)], the notched-noise method is based
on the phenomenon of masking. Masking is the process by which one sound (the "masker")
can interfere with the perception of another sound (the "signal").
The notched-noise method
consists of adjusting the spectrum level of a band-reject ("notched") noise masker to a
threshold at which a listener fails to detect a fixed-level pure-tone signal, as a function of the
width of the rejection band and its placement with respect to the tone.
The frequency-
resolving power of the auditory periphery is modeled as a bank of "auditory filters", whose
parameters are then estimated using the assumptions ("power spectrum model of masking",
Fletcher 1940) that (1) when detecting a signal in the presence of a masking sound, the
listener attends to the filter with the highest signal-to-noise power ratio at its output, (2)
detection thresholds correspond to a constant signal-to-masker power ratio at the output of
the filter, and (3) the filter is linear for the range of signal and masker levels used to define it.
In our physiological experiment, cochlear auditory filters were fit to thresholds for pure tones
in band-reject noise measured from single auditory nerve fibers using a paradigm adapted
from the one used in psychophysical
experiments.
A neural population model was then
developed to test the extent to which auditory filters estimated psychophysically match the
underlying
neural filters.
The similarity
observed
between the frequency
measured physiologically and the predicted psychophysical
selectivity
tuning supports the use of the
notched-noise method in human psychophysics. Despite the highly nonlinear nature of the
cochlea, estimates of neural frequency selectivity derived using the notched-noise method
show an increase in tuning sharpness with CF similar to that obtained in previous studies
using a variety of different stimuli and techniques and in broad agreement with that inferred
from results related to harmonic resolvability in Chapter 1.
10
Chapter
1
Pitch of Complex Tones: Rate-Place and Interspike-Interval
Representations in the Auditory Nerve
The work described in this chapter is published in the Journal of Neurophysiology:
Cedolin L and Delgutte B.
Pitch of Complex Tones: Rate-Place and Interspike Interval
Representations in the Auditory Nerve. J Neurophysiol 94: 347-362, 2005.
Reproduced with permission of the American Physiological Society.
11
1.1 ABSTRACT
Harmonic complex tones elicit a pitch sensation at their fundamental frequency (F0),
even when their spectrum contains no energy at F0, a phenomenon known as "pitch of the
missing fundamental".
individual
harmonics
The strength of this pitch percept depends upon the degree to which
are spaced sufficiently apart to be "resolved" by the mechanical
frequency analysis in the cochlea. We investigated the resolvability of harmonics of missingfundamental complex tones in the auditory nerve (AN) of anesthetized cats at low and
moderate stimulus levels, and compared the effectiveness of two representations
over a much wider range of FOs (110-3520 Hz) than in previous studies.
of pitch
We found that
individual harmonics are increasingly well resolved in rate responses of AN fibers as the
characteristic frequency (CF) increases. We obtained rate-based estimates of pitch dependent
upon harmonic resolvability by matching harmonic templates to profiles of average discharge
rate against CF.
These estimates were most accurate for FOs above 400-500 Hz, where
harmonics were sufficiently resolved.
We also derived pitch estimates from all-order
interspike-interval distributions, pooled over our entire sample of fibers. Such interval-based
pitch estimates, which are dependent upon phase-locking to the harmonics, were accurate for
FOs below 1300 Hz, consistent with the upper limit of the pitch of the missing fundamental in
humans. The two pitch representations are complementary with respect to the F0 range over
which they are effective; however, neither is entirely satisfactory in accounting for human
psychophysical data.
12
1.2 INTRODUCTION
A harmonic complex tone is a sound consisting of frequency components that are all
integer multiples of a common fundamental (F0). The pitch elicited by a harmonic complex
tone is normally very close to that of a pure tone at the fundamental frequency, even when
the stimulus spectrum contains no energy at that frequency, a phenomenon known as "pitch
of the missing fundamental".
Investigating the neural mechanisms underlying the perception of the pitch of harmonic
complex tones is of great importance for a variety of reasons. Changes in pitch convey
melody in music, and the superposition of different pitches is the basis for harmony. Pitch
has an important role in speech, where it carries prosodic features and information about
speaker identity.
In tone languages such as Mandarin Chinese, pitch also cues lexical
contrasts. Pitch plays a major role in auditory scene analysis: differences in pitch are a major
cue for sound source segregation, while frequency components that share a common
fundamental tend to be grouped into a single auditory object (Bregman 1990; Darwin and
Carlyon 1995).
Pitch perception with missing-fundamental stimuli is not unique to humans, but also
occurs in birds (Cynx and Shapiro 1986) and non-human mammals (Heffner and Whitfield
1976; Tomlinson and Schwartz 1988), making animal models suitable for studying neural
representations of pitch.
Pitch perception mechanisms in animals may play a role in
processing conspecific vocalizations, which often contain harmonic complex tones.
The neural mechanisms underlying pitch perception of harmonic complex tones have
been at the center of a debate among scientists for over a century (Seebeck 1841; Ohm 1843).
This debate arises because the peripheral auditory system provides two types of cues to the
pitch of complex tones: place cues dependent upon the frequency selectivity and tonotopic
mapping of the cochlea, and temporal cues dependent on neural phase locking.
The peripheral auditory system can be thought of as containing a bank of bandpass filters
representing the mechanical frequency analysis performed by the basilar membrane. When
two partials of a complex tone are spaced sufficiently apart relative to the auditory filter
bandwidths, each of them produces an individual local maximum in the spatial pattern of
basilar membrane motion.
In this case, the two harmonics are said to be "resolved" by the
13
auditory periphery.
On the other hand, when two or more harmonics fall within the pass-
band of a single peripheral filter, they are said to be "unresolved".
Because the bandwidths
of the auditory filters increase with their center frequency, only low-order harmonics are
resolved. Based on psychophysical data, the first 6-10 harmonics are thought to be resolved
in humans (Bernstein and Oxenham 2003b; Plomp 1964).
When a complex
tone contains resolved harmonics,
its pitch can be extracted by
matching the pattern of activity across a tonotopic neural map to internally stored harmonic
templates (Cohen et al. 1994; Goldstein 1973; Terhardt 1974; Wightman 1973). This type of
model accounts for many pitch phenomena, including the pitch of the missing fundamental,
the pitch shift associated with inharmonic complexes and the pitch ambiguity of complex
tones comprising only a few harmonics.
nature
of the neural
representation
However, a key issue in these models is the exact
upon
which the hypothetical
template-matching
mechanism operates.
Pitch percepts can also be produced by complex tones consisting entirely of unresolved
harmonics. In general, though, these pitches are weaker and more dependent on phase
relationships among the partials than the pitch based on resolved harmonics (Bernstein and
Oxenham 2003b; Carlyon and Shackleton 1994; Houtsma and Smurzynski 1990). With
unresolved harmonics, there are no spectral cues to pitch and therefore harmonic template
models are not applicable. On the other hand, unresolved harmonics produce direct temporal
cues to pitch because the waveform of a combination of unresolved harmonics has a period
equal to that of the complex tone. These periodicity cues, which are reflected in neural phase
locking, can be extracted by an autocorrelation-type mechanism (Licklider 1951; Meddis and
Hewitt 1991; Moore 1990; Yost 1996), which is mathematically equivalent to an all-order
interspike interval distribution for neural spike trains. The autocorrelation model also works
with resolved harmonics, since the period of the F is always an integer multiple of the
period of any of the harmonics; this common period can be extracted by combining (e.g.
summing) autocorrelation functions from frequency channels tuned to different resolved
harmonics (Meddis and Hewitt 1991; Moore 1990).
Previous neurophysiological studies of the coding of the pitch of complex tones in the
auditory nerve and cochlear nucleus have documented a robust temporal representation based
on pooled interspike interval distributions obtained by summing the interval distributions
14
from neurons covering a wide range of characteristic frequencies (Cariani and Delgutte
1996a,b; Palmer
representation
1990; Palmer and Winter
1993; Rhode
1995; Shofner
1991). This
accounts for a wide variety of pitch phenomena, such as the pitch of the
missing fundamental, the pitch shift of inharmonic tones, pitch ambiguity, the pitch
equivalence of stimuli with similar periodicity, the relative phase invariance of pitch and, to
some extent, the dominance of low-frequency harmonics in pitch. Despite its remarkable
effectiveness, the autocorrelation model has difficulty in accounting for the greater pitch
salience of stimuli containing resolved harmonics compared to stimuli consisting entirely of
unresolved harmonics (Bernstein and Oxenham 2003a; Carlyon 1998; Carlyon and
Shackleton 1994; Meddis and O'Mard 1997). This issue was not addressed in previous
physiological studies because they did not have a means of assessing whether individual
harmonics are resolved or not. Moreover, the upper F0 limit over which the interspikeinterval representation
of pitch is physiologically
viable has not been determined.
The
existence of such a limit is expected due to the degradation in neural phase locking with
increasing frequency (Johnson 1980).
In contrast to the wealth of data on the interspike-interval representation of pitch, possible
rate-place cues to pitch that might be available when individual harmonics are resolved by
the peripheral auditory system have rarely been investigated. The few studies that provide
relevant information (Hirahara et al. 1996; Sachs and Young 1979; Shamma 1985a,b) show
no evidence for rate-place cues to pitch, even at low stimulus levels where the limited
dynamic range of individual neurons is not an issue. The reason for this failure could be that
the stimuli used had low fundamental frequencies in the range of human voice (100-300 Hz)
and therefore produced few, if any, resolved harmonics in typical experimental animals,
which have a poorer cochlear frequency selectivity compared to humans (Shera et al. 2002).
Rate-place cues to pitch might be available in animals for complex tones with higher F0s in
the range of conspecific vocalizations, which corresponds to about 500-1000 Hz for cats
(Brown et al. 1978; Nicastro and Owren 2003; Shipley et al. 1991).
This hypothesis is
consistent with a report that up to 13 harmonics of a complex tone could be resolved in the
rate responses of high-CF units in the cat anteroventral cochlear nucleus (Smoorenburg and
Linschoten 1977).
15
In the present study, we investigated the resolvability of harmonics of complex tones in
the cat auditory nerve, and compared the effectiveness of rate-place and interval-based
representations of pitch over a much wider range of fundamental frequencies (110-3520 Hz)
than in previous studies. We found that the two representations
respect to the F
are complementary
with
range over which they are effective, but that neither representation is
entirely satisfactory in accounting for human psychophysical data. Preliminary reports of our
findings have been presented (Cedolin and Delgutte 2003, 2005a).
1.3 MATERIALS AND METHODS
Procedure
Methods for recording from auditory-nerve fibers in anesthetized cats were as described
by Kiang et al. (1965) and Cariani and Delgutte (1996a). Cats were anesthetized with Dial in
urethane (75 mg/kg), with supplementary doses given as needed to maintain an areflexic
state. The posterior portion of the skull was removed and the cerebellum retracted to expose
the auditory nerve. The tympanic bullae and the middle-ear cavities were opened to expose
the round
window.
Throughout
the experiment
the cat was given
injections
of
dexamethasone (0.26 mg/kg) to prevent brain swelling, and Ringer's solution (50 ml/day) to
prevent dehydration.
The cat was placed on a vibration-isolated table in an electrically-shielded,
temperature-
controlled, sound-proof chamber. A silver electrode was positioned at the round window to
record the compound action potential (CAP) in response to click stimuli, in order to assess
the condition and stability of cochlear function.
Sound was delivered to the cat's ear through a closed acoustic assembly driven by an
electrodynamic speaker (Realistic 40-1377). The acoustic system was calibrated to allow
accurate control over the sound-pressure level at the tympanic membrane.
Stimuli were
generated by a 16-bit digital-to-analog converter (Concurrent DA04H) using sampling rates
of 20 kHz or 50 kHz.
Stimuli were digitally filtered to compensate
for the transfer
characteristics of the acoustic system.
Spikes were recorded with glass micropipettes filled with 2 M KC1. The electrode was
inserted into the nerve and then mechanically advanced using a micropositioner (Kopf 650).
16
The electrode signal was bandpass filtered and fed to a custom spike detector. The times of
spike peaks were recorded with l-,us resolution and saved to disk for subsequent analysis.
A click stimulus at approximately 55 dB SPL was used to search for single units. Upon
contact with a fiber, a frequency tuning curve was measured by an automatic tracking
algorithm (Kiang et al. 1970) using 100-ms tone bursts, and the characteristic frequency (CF)
was determined. The spontaneous firing rate (SR) of the fiber was measured over an interval
of 20 s. The responses to complex-tone stimuli were then studied.
Complex-tone stimuli
Stimuli were harmonic complex tones whose fundamental frequency (F0) was stepped up
and down over a two-octave range. The harmonics of each complex tone were all of equal
amplitude, and the fundamental component was always missing. Depending on the fiber's
CF, one of four pre-synthesized stimuli covering different F0 ranges was selected so that
some of the harmonics would likely be resolved (Table 1). For example, for a fiber with a
1760 Hz CF, we typically used FOs ranging from 220 to 880 Hz so that the order of the
harmonic closest to the CF would vary from 2 to 8.
In each of the four stimuli, the
harmonics were restricted to a fixed frequency region as F0 varied (Table I). For each fiber,
the stimulus was selected so that the CF fell approximately at the center of the frequency
region spanned by the harmonics.
In some cases, data were collected from the same fiber in
response to two different stimuli whose harmonics spanned overlapping frequency ranges.
Each of the 50 F steps (25 up, 25 down) lasted 200 ms, including a 20-ms transition
period during which the waveform for one F0 gradually decayed while overlapping with the
gradual build up of the waveform for the subsequent F.
Spikes recorded during these
transition periods were not included in the analysis. Responses were typically collected over
20 repetitions of the 10-s stimulus (50 steps x 200 ms) with no interruption.
We used mostly low and moderate stimulus levels in order to minimize rate saturation,
which would prevent us from accurately assessing harmonic resolvability by the cochlea.
Specifically, the sound pressure level of each harmonic was initially set at 15-20 dB above
the fiber's threshold for a pure tone at CF, and ranged from 10 to 70 dB SPL with a median of
25 dB SPL. Since our stimuli contain many harmonics, overall stimulus levels are about 5-
17
10 dB higher than the level of each harmonic, depending on F0. In some cases, responses
were measured for two or more stimulus levels differing by 10-20 dB.
In order to compare neural responses to psychophysical data on the phase dependence of
pitch, three versions of each stimulus were generated with different phase relationships
among the harmonics: cosine phase, alternating (sine-cosine) phase, and negative Schroeder
phase
(Schroeder
1970).
The three
stimuli
have
the same power
spectrum
and
autocorrelation function, but differ in their temporal fine structure and envelope: while the
cosine-phase and alternating-phase stimuli have very "peaky" envelopes, the envelope of the
Schroeder-phase stimulus is nearly flat (Figure 1.1). Moreover, the envelope periodicity is at
F0 for the cosine-phase stimulus, but at 2xF0 for the alternating-phase stimulus. Alternatingphase stimuli have been widely used in previous studies of neural coding (Horst et al. 1990;
Palmer and Winter 1992, 1993).
Average-rateanalysis
For each step in the F0 sequence, spikes were counted over a 180-ms window extending
over the stimulus duration, but excluding the transition period between F steps. Spikes
counts from the two stimulus segments having the same F
descending parts of the F
(from the ascending and
sequence) were added together because response to both
directions were generally similar. The spike counts were converted to units of discharge rate
(spikes/s), and then plotted either as a function of F0 for a given fiber, or as a function of
fiber CF for a given F0 to form a "rate-place profile" (Sachs and Young 1979).
In order to assess the statistical reliability of these discharge rate estimates, "bootstrap"
resampling (Efron and Tibshirani 1993) was performed on the data recorded from each fiber.
One hundred resampled data sets were generated by drawing with replacement from the set
of spike trains in response to each F0. Spike counts in the ascending and descending part of
the F
sequence were drawn independently
from each other.
Spike counts from each
bootstrap data set were converted to discharge rate estimates as for the original data, and the
standard deviation of these estimates used as an error bar for the mean discharge rate.
Simple phenomenological
models were used to analyze average-rate responses to the
complex-tone stimuli. Specifically, a single-fiber model was fit to responses of a given fiber
as a function of stimulus F0 in order to quantify harmonic resolvability, while a population
18
model was used to estimate pitch from profiles of average discharge rate against CF for a
given F0.
The single-fiber model (Figure 1.2) is a cascade of three stages.
The linear bandpass
filtering stage, representing cochlear frequency selectivity, is implemented by a symmetric
rounded exponential function (Patterson 1976). The Sachs and Abbas (1974) model of ratelevel functions is then used to derive the mean discharge rate r from the r.m.s. amplitude p at
the ouput of the bandpass filter:
(1)
r = sp + rdm,,,x a +
p +
In this expression,
50
rp is the spontaneous rate, r,max is the maximum driven rate, and p50 is
the value of p for which the driven rate reaches half of its maximum value. The exponent a
was fixed at 1.77 to obtain a dynamic range of about 20 dB (Sachs and Abbas 1974). The
single-fiber model has a total of 5 free parameters: the center frequency and bandwidth of the
bandpass filter, rp, rdmw.,and pso. This number is considerably smaller than the 25 F0 values
for which responses were obtained in each fiber. The model was fit to the data by the least
squares method using the Levenberg-Marquardt algorithm as implemented by Matlab's
"lsqcurvefit" function.
The population model is an array of single-fiber models indexed on CF so as to predict
the auditory-nerve
rate response to any stimulus as a function of cochlear place.
The
population model has no free parameters; rather, it is used to find the stimulus parameters (F0
and SPL) most likely to have produced the measured rate-place profile, assuming that the
spike counts are statistically-independent
random variables with Poisson distributions whose
expected values are given by the model response at each CF.
The resulting maximum-
likelihood F estimate gives a rate-based estimate of pitch that does not require a priori
This strategy effectively implements the concept of
knowledge of the stimulus F.
"harmonic
template" used in pattern recognition models of pitch (Cohen et al. 1994;
Goldstein 1973; Terhardt 1974; Wightman 1973): here, the template is the model response to
a harmonic complex tone with equal amplitude harmonics.
In practice, the maximum-
likelihood F estimate was obtained by computing the model responses to complex tones
covering a wide range of F
in fine increments (0.1%) and finding the F
maximizes the likelihood of the data.
19
value that
While the population model has no free parameters, five fixed (i.e. stimulus-independent)
parameters still need to be specified for each fiber in the modeled population.
parameters
were selected
so as to meet two separate requirements:
These
(1) the model's
normalized driven rate must vary smoothly with CF, (2) the model must completely specify
the Poisson distribution of spike counts for each fiber so as to be able to apply the maximum-
likelihood method. To meet these requirements, three of the population-model parameters
were directly obtained from the corresponding parameters for the single-fiber model: the
center frequency of the bandpass filter (effectively the CF), the spontaneous rate r,, and the
maximum driven rate r,,,l(,. The sensitivity parameter P50so
in the population model was set to
the median value of this parameter over our fiber sample. Finally, the bandwidth of the
bandpass filter was derived from its center frequency by assuming a power law relationship
between the two (Shera et al. 2002). The parameters of this power function were obtained by
fitting a straight line in double logarithmic coordinates to a scatter plot of filter bandwidth
against center frequency for our sample of fibers.
Interspike-intervalanalysis
As in previous studies of the neural coding of pitch (Cariani and Delgutte 1996a, b;
Rhode 1995), we derived pitch estimates from pooled interspike interval distributions. The
pooled interval distribution is the sum of the all-order interspike interval distributions for all
the sampled auditory-nerve fibers, and is closely related to the summary autocorrelation in
the Meddis and Hewitt (1991) model.
The single-fiber interval distribution (bin width 0.1
ms) was computed for each F0 using spikes occurring in the same time window as used in
the rate analysis.
To derive pitch
estimates
from pooled
interval
distributions,
we used "periodic
templates" that select intervals at a given period and its multiples. Specifically, we define the
contrast ratio of a periodic template as the ratio of the weighted mean number of intervals for
bins within the template to the weighted mean number of intervals per bin in the entire
histogram. The estimated pitch period is the period of the template that maximizes the
contrast ratio. In computing the contrast ratio, each interval is weighted by an exponentially
decaying function of its length in order to give greater weight to short intervals.
This
weighting implements the idea that the lower F0 limit of pitch at about 30 Hz (Pressnitzer et
20
al. 2001) implies that the auditory system is unable to use very long intervals in forming
pitch percepts. A 3.6-ms decay time constant was found empirically to minimize the number
of octave and sub-octave errors in pitch estimation. The statistical reliability of the pitch
estimates was assessed by generating 100 bootstrap replications of the pooled interval
distribution (using the same resampling techniques as in the rate analysis), and computing a
pitch estimate for each bootstrap replication.
1.4 RESULTS
Our results are based on 122 measurements of responses to harmonic complex tones
recorded from 75 auditory-nerve fibers in two cats. Of these, 54 had high SR (> 18 spikes/s),
10 had low SR (< 0.5 spike/s), and 11 had medium SR. The CFs of the fibers ranged from
450 to 9200 Hz. We first describe the rate-responses of single fibers as a function of F to
characterize harmonic resolvability. We then derive pitch estimates from both rate-place
profiles and pooled interspike interval distributions, and quantify the accuracy and precision
of these estimates as a function of F0.
Single-fibercues to resolvedharmonics
Figure 1.3 shows the average discharge rate as a function of complex-tone F0 (harmonics
in cosine phase) for two auditory-nerve fibers with CFs of 952 Hz (panel A) and 4026 Hz
(panel B), respectively.
Data are plotted against the dimensionless ratio of fiber CF to
stimulus F, which we call harmonic number (lower horizontal axis). Because this ratio
varies inversely with F0, F0 increases from right to left along the upper axis in these plots.
The harmonic number takes an integer value when the CF coincides with one of the
harmonics of the stimulus, while it is an odd integer multiple of 0.5 (2.5, 3.5, etc...) when the
CF falls halfway between two harmonics. Thus, resolved harmonics should appear as peaks
in firing rate for integer values of the harmonic number, with valleys in between.
This
prediction is verified for both fibers at lower values of the harmonic number (higher F0s),
although, the oscillations are more pronounced and extend to higher harmonic numbers for
the high-CF fiber than for the low-CF fiber. This observation is consistent with the higher
21
quality factor (Q = CF/Bandwidth) of high-CF fibers compared to low-CF fibers (Kiang et al.
1965; Liberman 1978).
In order to quantify the range of harmonics that can be resolved by each fiber, a
simple peripheral auditory model was fit to the data (see Methods).
For both fibers, the
response of the best-fitting model (solid lines in Fig. 1.3) captures the oscillatory trend in the
data. The rate of decay of these oscillations is determined by the bandwidth of the bandpass
filter representing cochlear frequency selectivity in the model (Fig. 1.2). The harmonics of
F0 are considered to be resolved so long as the oscillations in the fitted curve exceed two
typical standard errors of the discharge rate obtained by bootstrapping (gray shading).
maximum resolved harmonic number
Nmax
The
is 4.1 for the low-CF fiber, smaller than Nmax for
the high-CF fiber (6.3). The ratio CF/Nmax gives F0min, the lowest fundamental frequency for
which harmonics are resolved in a fiber's rate response. In the examples of Fig. 1.3, F0min is
232 Hz for the low-CF fiber, and 639 Hz for the high-CF fiber.
Figure 1.4 shows how
Fmin
varies with CF for our entire sample of fibers.
To be
included in this plot, the variance of the residuals after fitting the single-fiber model to the
data had to be significantly smaller (p < 0.05, F-test) than the variance of the raw data so that
Nmax (and therefore F0min) could be reliably estimated.
were thus excluded;
Thirty five out of 122 measurements
23 of these had CFs below 2000 Hz.
On the other hand, the figure
includes data from fibers (shown by triangles) for which F0min was bounded by the lowest F0
presented and was therefore overestimated.
Fmin increases systematically with CF, and the
increase is well fit by a power function with an exponent of 0.63 (solid line). This increase is
consistent with the increase in tuning curve bandwidths with CF (Kiang et al. 1965).
Rate responses of AN fibers to complex stimuli are known to depend strongly on
stimulus level (Sachs and Young 1979). The representation of resolved harmonics in rate
responses is expected to degrade as the firing rates become saturated.
The increase in
cochlear filter bandwidths with level may further degrade harmonic resolvability.
We were
able to reliably fit single-fiber models to the rate-F0 data for stimulus levels as high as 38 dB
above the threshold at CF per component.
In general, this limit increased with CF, from
roughly 15 dB above threshold for CFs below 1 kHz to about 30 dB above threshold for CFs
above 5 kHz. No obvious dependence of this limit on fiber's spontaneous rate was noticed.
22
To more directly address the level dependence of responses, we held 24 fibers long
enough to record the responses to harmonic complex tones at two or more stimulus levels
differing by 10-20 dB.
In 23 of these 24 cases, the maximum resolved harmonic number
Nmaxdecreased with increasing level. One example is shown in Fig. 1.5 for a fiber with CF
at 1983 Hz. For this fiber, Nmax decreased from 7.1 at 20 dB SPL to 4.9 at 30 dB SPL.
The observed decrease in Nmax with level could reflect either broadened cochlear tuning
or rate saturation. To distinguish between these two hypotheses, two versions of the singlefiber model were compared when data were available at two stimulus levels. In one version,
all the model parameters were constrained to be the same at both levels; in the other version,
the bandwidth of the bandpass filter representing cochlear frequency selectivity was allowed
to vary with level.
The variable-bandwidth
model is guaranteed to fit the data better (in a
least squares sense) than the fixed-bandwidth
model because it has an additional
free
parameter. However an F-test for the ratio of the variances of the residuals revealed no
statistically significant difference between the two models at the 0.05 level for any of the 24
fibers, meaning than the additional free parameter of the variable-bandwidth
model gave it
only an insignificant advantage over the fixed-bandwidth model. This result suggests that
rate saturation, which is present in both models, may be the main factor responsible for the
decrease in Nmax with stimulus level.
Pitch estimationfrom rate-placeprofiles
Having characterized the limits of harmonic resolvability in rate responses of auditorynerve fibers, the next step is to determine how accurately pitch can be estimated from rate-
place cues to resolved harmonics. For this purpose, we fit harmonic templates to profiles of
average discharge rate against CF, and derive pitch estimates by the maximum likelihood
method, assuming
that the spike counts from each fiber are random variables
with
statistically-independent Poisson distributions. In our implementation, a harmonic template
is the response of a peripheral auditory model to a complex tone with equal-amplitude
harmonics.
The estimated pitch is therefore the F0 of the complex tone most likely to have
produced the observed response if the stimulus-response relationship were defined by the
model.
23
Figure 1.6 shows the normalized driven discharge rate of AN fibers as a function of CF in
response to two complex tones (harmonics in cosine phase) with FOs of 541.5 Hz (Panel A)
and 1564.4 Hz (Panel C). The rate is normalized by subtracting the spontaneous rate and
dividing by the maximum driven rate (Sachs and Young 1979), and these parameters are
estimated by fitting the single-fiber model to the rate-F0 data.
As for the single-fiber
responses in Fig. 1.3 and 1.5, responses are plotted against the dimensionless
harmonic
number CF/F0, with the difference that F is now fixed while CF varies, instead of the
opposite. Resolved harmonics should again result in peaks in firing rate at integer values of
the harmonic number. Despite considerable scatter in the data, this prediction is verified for
both FOs, although the oscillations are more pronounced for the higher F0. Many factors are
likely to contribute to the scatter, including the threshold differences among fibers with the
same CF (Liberman 1978), pooling data from two animals, intrinsic variability in neural
responses, and inaccuracies in estimating the minimum and maximum discharge rates used in
computing the normalized rate.
The solid lines in Fig. 1.6A and 1.6C show the normalized rate response of the population
model to the complex tone whose F0 maximizes the likelihood, i.e. the best fitting harmonic
template. Note that, while the figure shows the normalized model response, unnormalized
rates (actually, spike counts) are used when applying the maximum likelihood method
because only spike counts have the integer values required for a Poisson distribution. For
both FOs, the model response shows local maxima near integer values of the harmonic
number, indicating that the pitch estimates are very close to the stimulus FOs. This point is
shown more precisely in Fig. 1.6B and 1.6D, which show the log likelihood of the model
response as a function of template F0. Despite the very moderate number of data points in
the rate-place profiles, for both FOs the likelihood shows a sharp maximum when the
template F0 is very close to the stimulus F0. For the complex tone with F0 at 541.5 Hz, the
estimated pitch is 554.2 Hz, about 2% above the actual F.
For the 1564.4 Hz tone, the
estimated pitch is 1565.7 Hz, only 0.1% above the actual F0. The likelihood functions also
show secondary maxima for template FOs that form ratios of small integers (e.g. 4/3, 3/4)
with respect to the stimulus F0. However, because these secondary maxima are much lower
than the absolute maximum, the estimated pitch is highly unambiguous, consistent with
24
psychophysical observations for complex tones containing many harmonics (Houtsma and
Smurzynski 1990).
To assess the reliability of the maximum-likelihood pitch estimates, estimates were
computed for 100 bootstrap resamplings of the data for each F0 (see Method). Figure 1.7A
shows the median absolute estimation error of these bootstrap estimates as a function of F0
for complex tones with harmonics in cosine phase. With few exceptions, median pitch
estimates only deviate by a few percent from the stimulus F
above 500 Hz.
Larger
deviations are more common for lower F0s. The number and CF distribution of the fibers
had to meet certain constraints for each F0 to be included in the figure because, to reliably
estimate F,
the sampling of the CF axis has to be sufficiently dense to capture the
harmonically-related
oscillations in the rate-CF profiles.
This is why Fig. 1.7 shows no
estimates for F0s below 220 Hz and for a small subset of F0s (12 out of 56) above 220 Hz.
In order to quantify the salience of the rate-place cues to pitch as a function of F0, we
used the Fisher Information,
which is the expected value of the curvature of the log
likelihood function, evaluated at its maximum.
The expected value was approximated by
averaging the likelihood function over 100 bootstrap replications of the rate-place data. A
steep curvature means that the likelihood varies fast with template F0, and therefore that the
F0 estimate is very reliable. The Fisher Information was normalized by the number of data
points in the rate profile for each F0 to allow comparisons between data sets of different size.
Fig. 1.7B shows that the Fisher Information increases monotonically with F0 up to about
1000 Hz and then remains essentially constant.
Overall, pitch estimation from rate-place
profiles works best for F0s above 400-500 Hz, although reliable estimates were obtained for
F0s as low as 250 Hz.
Harmonic templates were fit to rate-place profiles obtained in response to complex tones
with harmonics in alternating phase and in Schroeder phase as well as in cosine phase in
order to test whether the pitch estimates depend on phase. Figure 1.8 shows an example for
an F0 of 392 Hz. The numbers of data points differ somewhat for the three phase conditions
because we could not always "hold" a unit sufficiently long to measure responses to all three
conditions.
Despite these sampling differences, the pitch estimates for the three phase
conditions are similar to each other (Panels A-C) and similar to the pitch estimate obtained
by combining data across all three phase conditions (Panel D).
25
We devised a statistical test for the effect of phase on pitch estimates. This test compares
the likelihoods of the rate-place data given two different models. In one model, the estimated
pitch is constrained to be the same for all three phase conditions by finding the F0 value that
maximizes the likelihood of the combined data (Fig. 1.8D). In the other model, a maximum
likelihood pitch estimate is obtained separately for each phase condition (Fig. 1.8A-C), and
then the maximum log likelihoods are summed over the three conditions, based on the
assumption that the 3 data sets are statistically-independent.
If the rate-place patterns for the
different phases differed appreciably, the maximum likelihood for the phase-dependent
model should be higher than that for the phase-independent model because the phasedependent model has the additional flexibility of fitting each data set separately. Contrary to
this expectation, when the two models were fit to 1000 bootstrap replications of the rateplace data, the distributions of the maximum likelihoods
for the two models did not
significantly differ (p = 0.178), indicating that the additional free parameters of the phasedependent model offer no significant advantage for this F0.
This test was performed for three different values of F0 (612 Hz, 670 Hz and 828 Hz) in
addition to the 392 Hz case shown in Fig. 1.81. In three of these four cases, the results were
as in Fig. 1.8 in that the differences in maximum likelihoods for the two models did not reach
statistical significance (p < 0.05).
For 612 Hz, the comparison did reach significance (p =
0.007), but for this F the rate-place profiles for harmonics in alternating and Schroeder
phase showed large gaps in the distribution of data points over harmonics numbers, making
the reliability of the F0-estimates for these two phases questionable. When the actual pitch
estimates for the different phase conditions were compared, there was no clear pattern to the
results across FOs, i.e. the pitch estimate for any given phase condition could be the largest in
one case and the smallest in another case.
These results indicate that phase relationships
among the partials of a complex tone do not seem to greatly influence the pitch estimated
from rate-place profiles, consistent with psychophysical data on the phase invariance of pitch
based on resolved harmonics (Houtsma and Smurzynski 1990).
'A programming error resulted in erroneous phase relationships among the harmonics for most FOs. We
only performed the test for the four FO values for which the phases were correct.
26
Pitch estimationfrom pooled interspikeintervaldistributions
Pitch estimates were derived from pooled interspike interval distributions to compare the
accuracy of these estimates with that of rate-place estimates for the same stimuli.
Figure
1.9A-B show pooled all-order interspike interval distributions for two complex-tone stimuli
with Fs
of 320 and 880 Hz (harmonics in cosine phase).
For both Fs,
the pooled
distributions show modes at the period of F0 and its integer multiples. However, these
modes are less prominent at the higher F0 for which only the first few harmonics are located
in the range of robust phase locking.
In previous work (Cariani and Delgutte 1996a,b; Palmer 1990; Palmer and Winter 1993),
the pitch period was estimated from the location of the largest maximum (mode) in the
pooled interval distribution. This simple method is also widely used in autocorrelation
models of pitch (Meddis and Hewitt 1991; Yost 1982, 1996). However, when tested over a
wide range of F0s, this method was found to yield severe pitch estimation errors for two
reasons.
First, for higher F0s as in Fig. 1.9B, the first interval mode near
/F0 is always
smaller than the modes at integer multiples of the fundamental period, due to the neural
relative refractory period.
Moreover, the location of the first mode is slightly but
systematically delayed with respect to the period of F0 (Fig. 1.9B), an effect also attributed to
refractoriness (McKinney and Delgutte 1999; Ohgushi 1983). In fact, a peak at the pitch
period is altogether lacking if the period is shorter than the absolute refractory period, about
0.6 ms for auditory-nerve fibers (McKinney and Delgutte 1999). These difficulties at higher
F0s might be overcome by using shuffled autocorrelograms (Louage et al. 2004) which,
unlike conventional autocorrelograms, are not distorted by neural refractoriness. However, a
more fundamental problem is that, at lower F0s, the modes at /F0 and its multiples all have
approximately the same height (Fig. 1.9A) so that, due to the intrinsic variability in neural
responses, some of the later modes will unavoidably be larger than the first mode in many
cases, and therefore lead to erroneous pitch estimates at integer submultiples of F0.
We therefore modified our pitch estimation method to make use of all pitch-related
modes in the pooled interval distribution rather than just the first one. Specifically, we used
periodic templates that select intervals at a given period and its multiples, and determined the
template F
which maximizes the contrast ratio, a signal-to-noise ratio measure of the
number of intervals within the template relative to the mean number of intervals per bin (see
27
Method). When computing the contrast ratio, short intervals were weighted more than long
intervals according to an exponentially decaying weighting function of interval length. This
weighting implements the psychophysical observation of a lower limit of pitch near 30 Hz
(Pressnitzer et al. 2001) by preventing long intervals to contribute significantly to pitch.
Fig. 1.9C-D show the template contrast ratio as a function of template FO for the same two
stimuli as on top. For both stimuli, the contrast ratio reaches an absolute maximum when the
template FO is very close to the stimulus FO, although the peak contrast ratio is larger for the
lower FO. The contrast ratio also shows local maxima one octave above and below the
stimulus FO. In Fig. 1.9C, these secondary maxima are small relative to the main peak at FO,
but in Fig. 1.9D the maximum at FO/2 is almost as large as the one at FO. Despite the close
call, FO was correctly estimated in both cases of Fig. 1.9 and, overall, our pitch estimation
algorithm produced essentially no octave or sub-octave errors over the entire range of FO
investigated (110-3520 Hz).
Figure 1.10 shows measures of the accuracy and strength of the interval-based pitch
estimates as a function of FO for harmonics in cosine phase.
The accuracy measure is the
median absolute value of the pitch estimation error over bootstrap replications of the pooled
interval distributions.
The estimates are highly accurate below 1300 Hz, where their medians
are within 1-2% of the stimulus FO (panel A).
pitch abruptly break down near 1300 Hz.
However, the interval-based estimates of
While the existence of such an upper limit is
consistent with the degradation in phase locking at high frequencies, the location of this limit
at 1300 Hz is low compared to the 4-5 kHz upper limit of phase locking, a point to which we
return in the Discussion.
Fig. 1.1OB shows a measure of the strength of the estimated pitch, the contrast ratio of the
best-fitting periodic template, as a function of FO. The contrast ratio is largest below 500 Hz,
then decreases gradually with increasing FO, to reach essentially unity (meaning a flat
interval distribution) at 1300 Hz. For FOs above 1300 Hz the modes in the pooled interval
distribution essentially disappear into the noise floor. Thus, the strength of interval-based
estimates of pitch is highest in the FO range where rate-based pitch estimates are the least
reliable due to the lack of strongly resolved harmonics. Conversely, rate-based estimates of
pitch become increasingly strong in the range of FOs where the interval-based estimates
break down.
28
For a few F0s, interval-based estimates of pitch were derived for complex tones with
harmonics in alternating phase and in Schroeder phase as well as for harmonics in cosine
phase. Figure 1.11 compares the pooled all-order interval distributions in the three phase
conditions for two F0s, 130 Hz (left) and 612 Hz (right). Based on the rate-place results, the
harmonics of the 130 Hz F0 are not resolved, while some of the harmonics of the 612 Hz F0
are resolved.
This is because we obtained a reliable pitch estimate based on rate-place
profiles at 612 Hz, but not at 130 Hz (Fig. 1.7).
For both F0s and all phase conditions, the distributions have modes at the period of the
fundamental frequency and its integer multiples (Fig. 1.11 A-C, E-G).
For the lower F0
(130 Hz), the pooled interval distribution for harmonics in alternating phase (panel B) also
shows secondary peaks at half the period of F0 and its odd multiples (arrows), reflecting the
periodicity of the stimulus envelope at 2xF0 (Fig 1.1).
Such frequency doubling has
previously been observed in auditory-nerve fiber responses to alternating-phase stimuli using
both period histograms (Horst et al. 1990; Palmer and Winter 1992) and autocorrelograms
(Palmer and Winter 1993).
At 130 Hz, the pooled interval distribution for harmonics in
negative Schroeder phase (panel C) shows pronounced modes at the period of F0 and its
multiples and strongly resembles the interval distribution for harmonics in cosine phase
(panel A), even though the waveforms of the two stimuli have markedly different envelopes
(Fig 1.1).
The interval-based pitch estimates are nearly identical for all three phase conditions, but
the maximum contrast ratio is substantially lower for harmonics in alternating phase than for
harmonics in cosine or in Schroeder phase (Fig. 1.11D).
In addition, for harmonics in
alternating phase, the contrast ratio of the periodic template at the envelope frequency 2xF0
is almost as large as the contrast ratio at F0. In contrast, for the higher F0 (612 Hz), there are
no obvious differences between phase conditions in the pooled all-order interval distributions
(Fig. 1.11E-G). In particular, the secondary peaks at half the period of F0, which were found
at 130 Hz for the alternating-phase
stimulus, are no longer present at 612 Hz. Moreover, the
maximum contrast ratios are essentially the same for all three phase conditions (Fig. 1.11H).
Overall, these results show that, while phase relationships among harmonics have little
effect on the pitch values estimated from pooled interval distributions, which are always
close to the stimulus F0, the salience of these estimates can be significantly affected by phase
29
when harmonics are unresolved. These results are consistent with psychophysical results
showing a greater effect of phase on pitch and pitch salience for stimuli consisting of
unresolved harmonics than for stimuli containing resolved harmonics (Houtsma and
Smurzynski 1990; Shackleton and Carlyon 1994). However, these results fail to account for
the observation that the dominant pitch is often heard at the envelope frequency 2xF0 for
unresolved harmonics in alternating phase.
1.5 DISCUSSION
Harmonicresolvabilityin auditory-nervefiberresponses
We examined the response of cat auditory-nerve fibers to complex tones with a missing
fundamental and equal-amplitude
harmonics.
We used low and moderate stimulus levels
(15-20 dB above threshold) to minimize rate saturation which would prevent us from
accurately assessing cochlear frequency selectivity and therefore harmonic resolvability from
rate responses.
In general, the average-rate of a single auditory-nerve
fiber was stronger
when its CF was near a low-order harmonic of a complex tone than when the CF fell halfway
in between two harmonics (Fig. 1.3).
This trend could be predicted using a
phenomenological model of single-fiber rate responses incorporating a bandpass filter
representing cochlear frequency selectivity (Fig. 1.2). The amplitude of the oscillations in
the response of the best-fitting single-fiber model, relative to the typical variability in the
data, gave an estimate of the lower F0 of complex tones whose harmonics are resolved at a
given CF (Fig. 1.3). This limit, which we call F0min, increases systematically with CF and
this increase is well fit by a power function with an exponent of 0.63 (Fig. 1.4). That the
exponent is less than 1 is consistent with the progressive sharpening of peripheral tuning with
increasing CF when expressed as a Q factor, the ratio CF/Bandwidth.
The exponent for Q
would be 0.37, which closely matches the 0.37 exponent found by Shera et al. (2002) for the
CF dependence of Q 0 in pure-tone tuning curves from AN fibers in the cat.
Our definition of the lower limit of resolvability F0min is to some extent arbitrary because
it depends on the variability in the average discharge rates, which in turn depends on the
number of stimulus repetitions and the duration of the stimulus. Nevertheless, our results are
30
consistent with those of Wilson and Evans (1971) for auditory-nerve fibers in the guinea pig
using ripple noise (a.k.a. comb-filtered noise), a stimulus with broad spectral maxima at
harmonically-related frequencies. These authors found that the number of such maxima that
can be resolved in the rate responses of single fibers (equivalent to our
CF from 2-3 at 200 Hz to about 10 at 10 kHz and above.
Nmax)
increases with
Similarly, Smoorenburg and
Linschoten (1977) report that the number of harmonics of a complex tone that are resolved in
the rate responses of single units in the cat anteroventral cochlear nucleus (AVCN) increases
from 2 at 250 Hz to 13 at 10 kHz. Despite the different metrics used to define resolvability,
both studies are in good agreement with the data of Fig. 1.4 if we use the conversion
FOmin
=
CF/Nnax.
Consistent with a previous report for AVCN neurons (Smoorenburg and Linschoten
1977), we found that the ability of auditory-nerve fibers to resolve harmonics in their rate
response degrades rapidly with increasing stimulus level (Fig. 1.5). This degradation could
be due either to the broadening of cochlear tuning with increasing level, or to saturation of
the average rate. Saturation seems to be the most likely explanation because a single-fiber
model with level-dependent bandwidth did not fit the data significantly better than a model
with fixed bandwidth. However, the level dependence of cochlear filter bandwidths might
have a greater effect on responses to complex tones if level were varied over a wider range
than the 10-20 dB used here (Cooper and Rhode 1997; Ruggero et al. 1997).
Rate-place representationof pitch
A major finding is that the pitch of complex tones could be reliably and accurately
estimated from rate-place profiles for fundamental frequencies above 400-500 Hz by fitting a
harmonic
template
to the data (Fig. 1.6 and 1.7A,B).
The harmonic
template
was
implemented as the response of a simple peripheral auditory model to a harmonic complex
tone with equal-amplitude harmonics, and the estimated pitch was the F0 of the complex tone
most likely to have produced the rate-place data assuming that the stimulus-response
relationship is characterized by the model. Despite the non-uniform sampling of CFs and the
moderate number of fibers sampled at each F0 (typically 20-40), these pitch estimates were
accurate within a few percent.
31
Pitch estimation became increasingly less reliable for FOs below 400-500, with large
estimation errors becoming increasingly common. Nevertheless, some reliable estimates
could be obtained for FOs as low as 250 Hz. This result is consistent with the failure of
previous studies to identify rate-place cues to pitch in auditory-nerve responses to harmonic
complex tones with FOs below 300 Hz (Hirahara et al. 1996; Sachs and Young 1979;
Shamma 1985a,b), although Hirahara et al. did find a weak representation of the first 2-3
harmonics in rate-place profiles for vowels with an F0 at 350 Hz.
In interpreting these results, it is important to keep in mind that the precision of the ratebased pitch estimates depends on many factors such as the number of fibers sampled, the CF
distribution of the fibers, pooling of data from two animals, the number of stimulus
repetitions, and the particular method for fitting harmonic templates.
For example, since the
lowest CF sampled was 450 Hz, the second harmonic and, in some cases, the third could not
be represented in the rate-place profiles for FOs below 220 Hz, possibly explaining why we
never obtained a reliable pitch estimate in that range. In fact, since our stimuli had missing
fundamentals, we cannot rule out that the fundamental might always be resolved when it is
present.
In one respect, our method may somewhat overestimate the accuracy of the rate-based
pitch estimates because we only included data from measurements
for which the rate
response as a function of F0 oscillated sufficiently to be able to reliably fit a single-fiber
model. This constraint was necessary because, for responses that do not oscillate, we could
not reliably estimate the minimum and maximum discharge rates which are essential in
fitting harmonic templates to the rate-place data. Thirty five of 122 responses were thus
excluded.
Because our design minimizes rate saturation, and because 23 or these 35
excluded responses were from fibers with CFs below 2 kHz, we infer that insufficient
frequency selectivity for resolving harmonics rather than rate saturation was the primary
reason for the lack of F0-related oscillations in these measurements.
A factor whose effect on pitch estimation performance is hard to evaluate is that the rateplace profiles included responses to stimuli presented at different sound levels. At first sight,
pooling data across levels might seem to increase response variability and therefore decrease
estimation performance.
However, because the stimulus level was usually selected to be 15-
20 dB above the threshold of each fiber so that responses would be robust without being
32
saturated, our procedure might actually have reduced the variability due to threshold
differences among fibers.
The rationale for this procedure is that an optimal central
processor would focus on unsaturated fibers because these fibers are the most informative.
Because level re. threshold rather than absolute level is the primary determinant of rate
responses, we are effectively invoking a form of the "selective listening hypothesis"
(Delgutte 1982; Delgutte 1987; Lai et al. 1994), according to which the central processor
attends to low-threshold,
high spontaneous rate fibers at low levels, and to high-threshold,
low-spontaneous fibers at high levels.
Our harmonic template differs from those typically used in pattern recognition models of
pitch in that it has very broad peaks at the harmonic frequencies.
Most pattern recognition
models (Duifhuis et al. 1982; Goldstein 1973; Terhardt 1974) use very narrow templates or
"sieves", typically a few percent of each harmonic's
frequency.
One exception is the
Wightman (1973) model, which effectively uses broad cosinusoidal templates by performing
a Fourier transform operation on the spectrum.
Our method also resembles the Wightman
model, and differs from the other models in that it avoids an intermediate, error-prone stage
which estimates the frequencies of the individual resolved harmonics; rather, a global
template is fit to the entire rate-place profile.
Broad templates are well adapted to the
measured rate-place profiles because the dips between the harmonics are often sharper than
the peaks at the harmonic frequencies (Figs. 6 and 8). On the other hand, the templates are
the response of the peripheral model to complex tones with equal-amplitude harmonics,
which exactly match the stimuli that were presented.
It remains to be seen how well such
templates would work when the spectral envelope of the stimulus is unknown, or when the
amplitudes of the individual harmonics are roved from trial to trial, conditions which cause
little degradation in psychophysical performance (Bernstein and Oxenham 2003a; Houtsma
and Smurzynski 1990).
Given the uncertainties about the various factors discussed above may affect our pitch
estimation procedure, a comparison of the pitch estimation performance with psychophysical
data should focus on robust overall trends as a function of stimulus parameters rather than on
absolute measures of performance. Both the precision of the pitch estimates (Fig. 1.7A) and
their salience (as measured by the Fisher information; Fig. 1.7B), improve with increasing F0
as the harmonics of the complex become increasingly resolved. This result is in agreement
33
with psychophysical observations that both pitch strength and pitch discrimination
performance improve as the degree of harmonic resolvability increases (Bernstein and
Oxenham 2003b; Carlyon and Shackleton
1994; Houtsma and Smurzynski 1990; Plomp
1967; Ritsma 1967). However, the continued increase in Fisher information with F0 beyond
1000 Hz conflicts with the existence of an upper limit to the pitch of missing-fundamental
stimuli, which occurs at about 1400 Hz in humans (Moore 1973b). This discrepancy between
the rapid degradation
in pitch discrimination
at high frequencies
and the lack of a
concomitant degradation in cochlear frequency selectivity is a general problem for place
models of pitch perception and frequency discrimination (Moore 1973a).
We also found that the relative phases of the resolved harmonics of a complex tone do
not greatly influence rate-based estimates of pitch (Fig. 1.8). This result is consistent with
expectations for a purely place representation of pitch, as well as with psychophysical results
for stimuli containing resolved harmonics (Houtsma and Smurzynski 1990; Shackleton and
Carlyon 1994; Wightman 1973).
The restriction of our data to low and moderate stimulus levels raises the question of
whether the rate-place representation of pitch would remain robust at the higher stimulus
levels typically used in speech communication or when listening to music. Previous studies
have used signal detection theory to quantitatively assess the ability of rate-place information
in the auditory nerve to account for behavioral performance
discrimination
(Viemeister
1988; Delgutte
in tasks such as intensity
1987; Winslow and Sachs 1988; Winter and
Palmer 1991; Colburn et al. 2003) and formant-frequency discrimination for vowels (Conley
and Keilson 1995; May et al. 1996). These studies give a mixed message. On the one hand,
the rate-place
representation
generally contains
sufficient
information
to account for
behavioral performance up to the highest sound levels tested. On the other hand, because the
fraction of high-threshold
fibers is small compared
to low-threshold
fibers, predicted
performance of optimal processor models degrades markedly with increasing level, while
psychophysical performance remains stable. Thus, while a rate-place representation cannot
be ruled out, it fails to account for a major trend in the psychophysical data. Extending this
type of analysis to pitch discrimination for harmonic complex tones is beyond the scope of
this paper.
Given the failure of the rate-place representation
to account for the level
dependence of performance in the other tasks, a more productive approach may be to explore
34
alternative spatio-temporal representations that would rely on harmonic resolvability like the
rate-place representation, but would be more robust with respect to level variations by
exploiting phase locking (Shamma, 1985a, Heinz et al. 2001). Preliminary tests of one such
spatio-temporal representation are encouraging (Cedolin and Delgutte 2005b).
Interspike-intervalrepresentationof pitch
Our results confirm previous findings (Cariani and Delgutte 1996a,b; Palmer 1990;
Palmer and Winter 1993), that fundamental frequencies of harmonic complex tones are
precisely represented in pooled all-order interspike-interval distributions of the auditory
nerve. These interval distributions have prominent modes at the period of F0 and its integer
multiples (Fig.
.9A, B).
Pitch estimates derived using periodic templates that select
intervals at a given period and its multiples were highly accurate (often within 1%) for F0s
up to 1300 Hz (Fig. 1.10).
The determination of this upper limit to the interval-based
representation of pitch is a new finding.
Moreover, the use of periodic templates for pitch
estimation improves upon the traditional method of picking the largest mode in the interval
distribution by greatly reducing sub-octave errors.
While the existence of an upper limit to the representation of pitch in interspike intervals
is expected from the degradation in phase locking at high frequencies, the location of this
limit at 1300 Hz is low compared to the usually quoted 4-5 kHz limit of phase locking in the
auditory nerve (Johnson
1980; Rose et al. 1967).
Of course, both the limit of pitch
representation and the limit of phase locking depend to some extent on the signal-to-noise
ratio of the data, which in turn depends on the duration of stimulus, the number of stimulus
repetitions and, for pooled interval distributions, the number of sampled fibers. However the
discrepancy between the two limits appears too large to be entirely accounted by differences
in signal-to-noise ratio. Fortunately, the discrepancy can be largely reconciled by taking into
account harmonic resolvability and the properties of our stimuli. For F0s near 1300 Hz, all
the harmonics within the CF range of our data (450-9200 Hz) are well resolved (Fig. 1.4), so
that information about pitch in pooled interval distributions must depend on phase locking to
individual resolved harmonics rather than on phase locking to the envelope generated by
interactions between harmonics within a cochlear filter. Moreover, because our stimuli have
missing fundamentals, an unambiguous determination of pitch from the pooled distribution
35
requires phase locking to at least two resolved harmonics.
As F0 increases above 1300 Hz,
the third harmonic (3900 Hz) begins to exceed the upper limit of phase locking, leaving only
ambiguous pitch information and therefore leading to severe estimation errors.
A major finding is that the range of F0s over which interval-based estimates of pitch are
reliable
roughly covers the entire human perceptual
range of the pitch of missing-
fundamental stimuli, which extends up to 1400 Hz for stimuli containing many harmonics
(Moore 1973b). It is widely recognized that the upper limit of phase locking to pure tones
matches the limit in listeners' ability to identify musical intervals (Semal and Demany 1990;
Ward 1954). The present results extend this correspondence to complex tones with missing
F0.
Our results predict that pitch based on pooled interval distributions is strongest for F0s
below 400 Hz (Fig. 1.10B), a range for which the decreased effectiveness of pitch estimation
based on rate-place information implies that individual harmonics are poorly resolved in the
cat. Thus, the interval-based representation of pitch seems to have trouble predicting the
greater salience of pitch based on resolved harmonics compared to that based on unresolved
harmonics (Shackleton and Carlyon 1994; Bernstein and Oxenham 2003b). This conclusion
based on physiological data supports similar conclusions previously reached for
autocorrelation models of pitch perception (Bernstein and Oxenham 2003a; Carlyon 1998;
Meddis and O'Mard 1997).
We found that neither the pitch values nor the pitch strength estimated from pooled
interval distributions depend on the phase relationships among the harmonics at higher F0s
where some harmonics are well resolved (Fig. 1.1lE-G).
This finding is consistent with
psychophysical observations on the phase invariance of pitch and pitch salience for stimuli
containing resolved harmonics (Carlyon and Shackleton 1994; Houtsma and Smurzynski
1990; Wightman 1973). However, pitch salience and, in some cases, pitch values do depend
on phase relationships for stimuli consisting of unresolved harmonics (Houtsma and
Smurzynski 1990; Lundeen and Small 1984; Ritsma and Engel 1964).
In particular, for
stimuli in alternating sine-cosine phase, the pitch often matches the envelope periodicity at
2xF0 rather than the fundamental (Lundeen and Small 1984).
Consistent with previous
studies (Horst et al. 1990; Palmer and Winter 1992; Palmer and Winter 1993), we found a
correlate of this observation in the pooled interval distributions in that our interval-based
36
measure of pitch strength was almost as large at the envelope frequency 2xF0 as at the
fundamental F0 for alternating-phase stimuli with unresolved harmonics (Fig 1ID). Despite
this frequency doubling, the pitch values estimated from interval distributions were always at
F0 and never at 2xF0, in contrast to psychophysical judgments for unresolved harmonics in
alternating phase (Lundeen and Small 1984). Thus, pitch estimation based on interspike
intervals does not seem to be sufficiently sensitive to the relative phases of unresolved
harmonics compared to psychophysical data. A similar conclusion has been reached for the
autocorrelation model of pitch, and modifications to the model have been proposed in part to
handle this difficulty (de Cheveign6 1998; Patterson and Holdsworth 1994).
Vocalizationsand pitch perception
A widely held view is that pitch perception for harmonic complex tones is closely linked
to the extraction of biologically relevant information from conspecific vocalizations
including human speech. For example, Terhardt (1974) argues: "The virtual pitch cues can
be generated only if a learning process previously has been performed. ... In that process,
the correlations between the spectral-pitch cues of voiced speech sounds ... are recognized
and stored. The knowledge about harmonic pitch relations which is acquired in this way is
employed by the system in the generation of virtual pitch."
While the role of a learning
mechanism may be questioned given that the perception of missing-fundamental pitch
appears to be already present in young infants (Clarkson and Clifton 1985; Montgomery and
Clarkson 1997), a link between vocalization and pitch perception is supported by other
arguments. Many vertebrate vocalizations contain harmonic complex tones such as the
vowels of human speech. Because the fundamental component is rarely the most intense
component in these vocalizations, it would often be masked in the presence of environmental
noise, thereby creating a selective pressure for a missing-fundamental mechanism. The link
between vocalization and pitch perception has recently been formalized using a model which
predicts many psychophysical pitch phenomena from the probability distribution of human
voiced speech sounds, without explicit reference to any specific pitch extraction mechanism
(Schwartz and Purves 2004).
In cats, the fundamental frequency range most important for vocalizations lies at 5001000 Hz (Brown et al. 1978; Nicastro and Owren 2003; Shipley et al. 1991). In this range,
37
pitch is robustly represented in both rate-place profiles and pooled interspike interval
distributions. The difficulties encountered by the rate-place representation over the F0region below 300 Hz, which is the most important for human voice, may reflect the poorer
frequency selectivity of the cat cochlea compared to the human (Shera et al. 2002). If we
assume that human cochlear filters are about three times as sharply tuned as cat filters,
consistent with the Shera et al. data, the rate-place representation would hold in humans for
FOs at least as low as 100 Hz, thereby encompassing most voiced speech sounds. Thus, in
humans as well as in cats, the pitch of most conspecific vocalizations may be robustly
Such a dual
represented in both rate-place profiles and interspike interval distributions.
representation
may be advantageous
in situations when either mechanism is degraded by
either cochlear damage or central disorders of temporal processing because it would still
allow impaired individual to extract pitch information.
Despite its appeal, the idea that the pitch of conspecific
vocalizations
has a dual
representation in spatial and temporal codes is not likely to hold across vertebrate species. At
one extreme is the mustached bat, where the FOs of vocalizations range from 8 to 30 kHz
(Kanwal et al. 1994), virtually ruling out any temporal mechanism.
Evidence for a
perception of the pitch of missing fundamental stimuli at ultrasonic frequencies is available
for one species of bats (Preisler and Schmidt 1998). At the other extreme is the bullfrog,
where the fundamental frequency of vocalizations near 100 Hz appears to be coded in the
phase locking of auditory-nerve
fibers to the sound's envelope rather than by a place
mechanism (Dear et al. 1993; Simmons et al. 1990). Although this species is sensitive to the
fundamental frequency of complex tones (Capranica and Moffat 1975), it is not known
whether it experiences a missing-fundamental phenomenon similar to that in humans. These
examples suggest that a tight link between pitch and vocalization may be incompatible with
the existence of a general pitch mechanism common to all vertebrate
species.
Either
different species use separate pitch mechanisms to different degrees, or the primary function
of the pitch mechanism is not to extract information from conspecific vocalizations, or both.
38
1.6 CONCLUSIONS
We compared the effectiveness of two possible representations of the pitch of harmonic
complex tones in the responses of the population of auditory-nerve
fibers at low and
moderate stimulus levels: a rate-place representation based on resolved harmonics and a
temporal representation based on pooled interspike-interval distributions. A major finding is
that the rate-place representation was most effective for F0s above 400-500 Hz, consistent
with previous reports of a lack of rate-place cues to pitch for lower Fs, and with the
improvement in cochlear frequency selectivity with increasing frequency. The interspikeinterval representation gave precise estimates of pitch for low F0s, but broke down near 1300
Hz. This upper limit is consistent with the psychophysical limit of the pitch of the missing
fundamental for stimuli containing many harmonics, extending to missing-fundamental
stimuli the correspondence between the frequency range of phase locking and that of musical
pitch.
Both rate-place and interspike-interval
representations were effective in the F0 range
of cat vocalizations, and a similar result may hold for human voice if we take into account
the differences in cochlear frequency selectivity between the two species. Consistent with
psychophysical data, neither of the two pitch representations was sensitive to the relative
phases of the partials for stimuli containing resolved harmonics.
On the other hand, neither representation of pitch is entirely consistent with the
psychophysical data. The rate-place representation fails to account for the upper limit of
musical pitch and is known to degrade rapidly with increases in sound level and decreases in
signal-to-noise
ratio.
The interval representation has trouble accounting for the greater
salience of pitch based on resolved harmonics compared to pitch based on unresolved
harmonics and appears to be insufficiently sensitive to phase for stimuli consisting of
unresolved harmonics. These conclusions suggest a search for alternative neural codes for
pitch that would combine some of the features of place and temporal codes in order to
overcome
the limitations
of either code.
One class of codes that may meet these
requirements are spatio-temporal codes which depend on both harmonic resolvability and
39
phase locking (Loeb et al. 1983; Shamma and Klein 2000; Shamma 1985a; Heinz et al.
2001).
1.7 ACKNOWLEDGMENTS
The authors would like to thank Christophe Micheyl, Christopher Shera, and Joshua
Bernstein for valuable comments on the manuscript and Andrew Oxenham for his advice on
the design of the experiments. Thanks also to Connie Miller for her expert surgical
preparation of the animals. This work was supported by NIH grants RO1 DC02258 and P30
DC05209.
40
Fixed stimulus
CF Range (Hz)
FO Range (Hz)
440-1760
110-440
440-1760
880-3520
220-880
880-3520
1760-7040
440-1760
1760-7040
3520-14080
880-3520
3520-14080
frequencyregion(Hz)
Table 1. Parameters of the four complex-tone stimuli with varying FO, and range of CFs for
which each stimulus was used.
41
Cosine Phase
Ae
I
Alternating Sine-Cosine Phase
u
E
L.
7
_
en
Negative Schroeder Phase
0
3
2
1
Time (fractions of stimulus period)
4
Figure 1.1: different phase relationships among the harmonics give rise to different stimulus
waveforms. For harmonics in cosine phase (top), the waveform shows one peak per period of
the FO. When the harmonics are in alternating phase (middle), the waveform peaks twice
every period of the FO. A negative Schroeder phase relationship among the harmonics
(bottom) minimizes the amplitude of the oscillations of the envelope of the waveform.
42
Amplitude
Compression
Ro-Exp
Bandpass Filter
I
I
r
Sound
-
I
Pressure
*1
I
rms
Amplitude
J/.max
rsp........
P50
I
Average
Rate
dB
Figure 1.2. Single-fiber average-rate model. The first stage represents cochlear frequency
selectivity, implemented by a symmetric rounded-exponential bandpass filter (Patterson
1977). The Sachs and Abbas (1974) model of rate-level functions is used to compute the
mean discharge rate from the r.m.s. amplitude at the ouput of the bandpass filter.
43
A
Fundamental
476
318
RESOLVED
...ca
0,)
Frequency
191
238
159
(Hz)
136
119
106
UNRESOLVED
65
0::
'0
0,)
0')0,)
~
f/)
ca_
.c:
f/)
0,)
f/).:JJ:.
<J
60
55
C'a.
c ~
ca
50
0,)
~
45
CF
2
3
4
5
Harmonic
B
6
1342
1006
8
Frequency
805
671
9
(Hz)
575
503
447
UNRESOLVED
120
...ca
7
Hz
Number CF/FO
Fundamental
2012
= 952
0,)
0::
100
0,)-
O')<J
~
0,)
caJE.
.c:
f/)
0,)
f/).:JJ:.
•
80
<J
c'a.
c~
60
ca
0,)
~
40
CF
2
3
= 4026
4
5
Harmonic
Figure 1.3. Average discharge
Hz
6
7
rate against complex-tone
same cat with CFs of 952 Hz (A) and 4026 Hz (B).
harmonic
number CF/FO, FO increases
with errorbars
resampling
rate
:t 1
of the stimulus trials (see Method).
FO for two AN fibers from the
Because
the lower axis shows the
standard
deviation
The intersection
from the upper envelope
obtained
Filled circles
by bootstrap
The solid lines show the response of the best-
model (Fig. 1,2). The upper and lower envelopes
shown by dotted lines.
deviations
9
from right to left along the upper axis,
show the mean discharge
fitting single-fiber
8
Number CF/FO
of the lower envelope
(gray shading)
N"un for which harmonics are resolved (vertical lines).
44
of the fitted curve are
with two typical standard
gives the maximum
harmonic
number
2000
N
x
1000
0
I..,
oLL
>
7
._
E
U) 0
4) Lo
U)
n)
3
0
inn
400
1000
10000
Characteristic Frequency (Hz)
Figure 1.4. Lowest resolved F as a function of characteristic frequency. Each point shows
data from one auditory-nerve fiber. Triangles show data point for which FOminwas somewhat
overestimated because harmonics were still resolved for the lowest F presented. The solid
line shows the best-fitting straight line on double logarithmic coordinates (a power law).
45
Fundamental
180
661
496
Frequency (Hz)
397
330
CF
Resolved at
30 dB SPL
160
-
283
= 1983
248
Hz
•
Q)
140
m
a:
Q)
E>o
m Q)
120
.c en
0"""
en
en
.-
o~
C
100
Q)
Q.
m~
Q)
~
80
Resolved at
20 dB SPL
60 0
•
40
345
678
Harmonic Number CF/FO
Figure
1.5.
auditory-nerve
Effect
of stimulus
level on harmonic
fiber (CF = 1983 Hz).
against FO for complex
Open and filled circles
tones at 20 and 30 dB SPL, respectively.
of the best fitting model when model parameters
stimulus
resolvability
levels. Other features as in Fig. 1.3.
46
were constrained
in rate responses
show mean discharge
of an
rate
Solid lines show response
to be the same for both
A
Characteristic Frequency (Hz)
C
Characteristic Frequency (Hz)
3128
4692
6256
7820
1
0.8
0.6
o:
0.4
0.2
0
3
4
5
6
7
8
1.5
Harmonic Number (CFIFO)
.0,
Z
D
3
-200
Template
271
D
FO (Hz)
542
1084
-400
-1500
z._
-600
-2000
n oO
IL
3
3.5
4
4.5
Template
FO (Hz)
1564
3128
-1000
E-
u->
2.5
782
o
.o
2
Harmonic Number (CF/FO)
-2500
-800
-3000
0.5
Template
1
FO Stimulus
2
1.A,
0.5
Template
FO
1
FO Stimulus
2
FO
Figure 1.6. Maximum-likelihood pitch estimation from rate-place profiles using harmonic
templates for two complex tones with FOs at 542 Hz (A, B) and 1564 Hz (C, D), respectively.
A and C: Filled circles show normalized driven rate as a function of both CF (upper axis) and
harmonic number CF/FO (lower axis). Solid lines show the maximum-likelihood harmonic
template, which is the response of a population model to a complex tone with equal-
amplitude harmonics (see Method). B and D: Log likelihood of the harmonic template model
in producing the data points as a function of template FO.
47
5
10
A
i' 8
8A
.60
6
E
1
· ·
IA
go
~~~~4b
2
4000
1000
100
~B
0**
4.00
W(
NE
0 10
.
.
I10 3 .0.0
~
1000
100
4000
Fundamental Frequency (Hz)
Figure 1.7. Pitch estimation based on rate-place profiles: median absolute estimation error
(A) and normalized Fisher Information (B) as a function of FO. The median is obtained over
100 bootstrap resamplings of the data (see Method). Triangles indicate data points for which
the median was out of the range defined by the vertical axes. The Fisher Information is a
measure of pitch strength defined as the curvature of the log likelihood with respect to
template FO at the location of the maximum.
48
Cosine Phase
Alternating Phase
Characteristic
A
784
"C (I)
(1)-
m c
E... >
0.6
.-N [[CU
B
2352
o
0
0.8
1568
Frequency
784
0
1568
y
2352
y
y
yy
\/\i\J'7\'
(I)
O.t:
ZC
(Hz)
0.4
0.2
~
0
2
6
4
yy
2
Harmonic
6
4
Number (CF/FO)
FO = 392 Hz
Schroeder Phase
All Phases
Characteristic
C
1568
784
o
"C
Frequency
0
2352
(Hz)
2352
784
00 0
iJ
~o
y
(I)
(1)-
.- [[
N
CU
o
m
c
E
(I)
... >
00
~
0
00
00
y
.,
0
•
rWy
..
'j
\)
\~
o ogo
o
o
~o
o
2
6
4
2
Harmonic
place profiles.
Stimulus
likelihood
harmonic
alternating
(triangles)
and Schroeder
and maximum
likelihood
on pitch estimation
FO = 392 Hz. A,B and C: normalized
maximum
templates
harmonic
(black
(circles)
lines)
(gray
likelihood
across phases is plotted in gray also in panels A-C.
49
driven
for harmonics
phase, respectively.
template
as in panels A-C). The maximum
6
4
Number (CF/FO)
Figure 1.8. Effect of the relative phase of the harmonics
(symbols
0 0
'6>
o
o.t:
zc
0
---12b
Tl9'
line)
rate (symbols)
in cosine
D: normalized
for data
harmonic
based on rate-
pooled
template
and
(squares),
driven rate
across
phases
for data pooled
F0 = 320 Hz
A
1200
1200
0
1000
1000
1-
800
800
= c
600
M
E
zT.
O
I
I
l
600
II1
400
200
200
I
400
0
0
5
10
15
III III III III III Ili
l
i-
l
.
llJ.
.l..l
._._.
.
.
l..l
A JILM
II-,
l
.1
0
25
20
0
Interspike Interval (msec)
Template
F0 = 880 Hz
B
5
10
15
20
25
Interspike Interval (msec)
FO (Hz)
Template
440
3
FO (Hz)
880
1760
0O
.= 2
2
o el 1
1
I...
.
.
Oo
0
0
0.5
Template
1
FO Stimulus
0
2
0.5
Template
FO
1
FO Stimulus
2
FO
Figure 1.9. Pitch estimation based on pooled all-order interspike interval distributions for
two FOs, 320 Hz
(A,C) and 880 Hz (B,D). A, B: Pooled all-order interspike interval
distributions with periodic templates maximizing the contrast ratio (vertical dotted lines). C,
D: periodic template contrast ratio as a function of its F0.
50
10
_A
A
'~,..
.& _- 8
as'-
.0W
6
r2 4
Co
X IE
2
1
1^
~
w 0 1oW
100
4
3.5-
4)
0
~
0 oo4o
1000
4000
Be
a)@
2.5
X
'
4) O
M
2
1.5
1
100
.
1000
FundamentalFrequency
4000
(Hz)
Figure 1.10. Pitch estimation based on pooled all-order interspike interval distributions:
median absolute estimation error and best- template "contrast ratio" (see Methods) as a
function of FO, Panel A: median (over 100 bootstrap resampling trials) pitch absolute
estimation error, expressed as percentage of the true FO. Triangles: the pitch absolute
estimation error exceeded 10% of the true FO. Panel B: contrast ratio of the best-matching
periodic template.
51
FO = 130 Hz
FO=612Hz
E
A
Cosine
Phase
....
ev
.0
E
::J
en
z~
Alternating
Phase
ev >
....
N ev
"C
='E
E-0
....
~-
o
z
Schroeder
Phase
o
5
10
15
20
o
25
5
Interspike Interval (msec)
Template FO(Hz)
3
ev
120..-0
E ro
ev[[
2.5
~
~
1.5
"C
....
130
10
15
20
25
Interspike Interval (msec)
260
Template FO(Hz)
H612
D
1224
3
-
Cosine Phase
Alternating Phase
.--... Schroeder Phase
-
Cosine Phase
Alternating Phase
...... Schroeder Phase
2.5
2
o'E0
.t:;
cf
U
0.5
0.5
o
o
2
2
Template FOI Stimulus FO
Figure 1.11. Effect of the relative
pooled all-order interval distributions
612 Hz in E-H).
Template FOI Stimulus FO
phase of the harmonics
for two fundamental
A-E: Pooled interval distributions
alternating
sine-cosine
phase
(B,F)
and Schroeder
secondary
local maxima
frequency)
of the stimulus FO and its odd multiples.
in the interval
distribution
a function of its FO for the three phase conditions.
52
on pitch estimation
frequencies
based on
(130 Hz in A-D, and
for harmonics
in cosine phase (A, E),
phase
Arrows
(C,G).
at half the period
0, H: Periodic
in B point
(i.e. twice
to
the
template contrast ratio as
Chapter 2
Spatio-Temporal Representation
of the Pitch of Complex Tones in the Auditory Nerve
53
2.1 INTRODUCTION
In Chapter
(Cedolin and Delgutte 2005c), we tested the effectiveness of two classic
representations of the pitch of harmonic complex tones at the level of the auditory nerve in
the cat:
a rate-place
representation
based
representation
on
pooled
based on
resolved
interspike-interval
harmonics
distributions.
and
a temporal
The
rate-place
representation was most effective for fundamental frequencies (F0s) above 400-500 Hz,
while the interspike-interval
broke down near 1300 Hz.
effective in the F
representation gave precise estimates of pitch for low F0s, but
Both rate-place and interspike-interval
range of cat vocalizations,
representations
were
which corresponds to about 500-1000 Hz
(Brown et al. 1978; Nicastro and Owren 2003; Shipley et al. 1991). Neither of the two pitch
representations was sensitive to the relative phases of the partials for stimuli containing
resolved harmonics, consistent with human psychophysical data (Houtsma and Smurzynski
1990; Shackleton and Carlyon 1994).
Neither representation of pitch was entirely consistent with available human
psychophysical data. The strength of the rate-place representation was predicted to increase
monotonically with F0, thus failing to predict the existence of an upper F0-limit for the pitch
of missing-fundamental
complex tones. The rate-place representation also degrades rapidly
with increases in sound level and decreases in signal-to-noise
ratio, in contrast to the
relatively robust representation suggested by human psychophysics.
The interval
representation had trouble accounting for the greater salience of pitch based on resolved
harmonics
compared
to pitch based on unresolved
harmonics
and appeared
to be
insufficiently sensitive to phase for stimuli consisting of unresolved harmonics.
The present study was aimed at investigating an alternative, "spatio-temporal"
neural
representation of pitch, which might combine the advantages and overcome the limitations of
the rate-place and interval representations.
Spatio-temporalrepresentationof pitch
The variation with cochlear place of the phase of basilar membrane motion in response to
a pure tone is fastest in proximity of the cochlear place tuned to the tone frequency
(Anderson 1971; Pfeiffer and Kim 1975; van der Heijden and Joris 2003). At frequencies
54
below the limit of phase-locking, this rapid spatial change in phase is reflected in the timing
of auditory-nerve firings, thus generating a cue to the frequency of the pure tone (Shamma
1985a).
Our hypothesis is that rapid changes in phase should occur at each of the spatial locations
tuned to a resolved harmonic.
Hence, "spatio-temporal"
cues to the frequencies of each of
the resolved harmonics of a complex tone would be provided by the wave of excitation
produced in the cochlea by each cycle of the stimulus, as the wave travels from the base to
the apex.
We illustrate
physiologically-realistic
this concept
peripheral
by showing
(Figure 2.1) the response
of a
auditory model (Zhang et al., 2001) to a missing-
fundamental harmonic complex tone with a 200 Hz F0. In this example, the bandwidths of
the peripheral filters were adjusted to fit human psychophysical masking data (Glasberg and
Moore 1990). Each cycle of the stimulus waveform creates a wave of excitation traveling
from the base (high CFs) to the apex (low CFs) of the cochlea. The velocity of the traveling
wave is not uniform, but varies such that the response latency changes more rapidly for CFs
near a resolved harmonic of the complex tone than for CFs in between two harmonics.
In principle., these latency cues can be extracted by a central neural mechanism sensitive
to the relative timing of spikes in AN fibers innervating neighboring cochlear locations,
effectively converting spatio-temporal cues into rate-place cues. One possible mechanism is
lateral inhibition, whose effect would likely be increments in discharge rate as inputs with
neighboring CFs become less coincident (as would be the case when the phase change with
CF is fast). Such mechanism would perform the equivalent of a derivative with respect to
space of the AN spatio-temporal response pattern (Shamma, 1985b).
To simulate
the extraction
of these spatio-temporal
cues by a lateral inhibition
mechanism, we compute the spatial derivative of the response pattern (a point by point
difference between adjacent rows in the image of Fig. 2.1), then integrate the absolute value
of the derivative over time. This "mean absolute spatial derivative" shows local maxima at
CFs corresponding to the frequencies of harmonics 2-6. The average discharge rate, on the
other hand, is largely saturated at this stimulus level.
Thus, according to our model simulation, spatio-temporal pitch cues should be available
in the spatio-temporal response pattern of AN fibers, and can, in principle, be decoded by a
lateral inhibition mechanism. Moreover, spatio-temporal cues to resolved harmonics may
55
persist at stimulus levels at which the rate-place representation fails due to the saturation of
individual fibers average-rate response.
Scaling Invariance
To accurately measure the entire spatio-temporal response pattern of the AN for a
complex-tone stimulus composed of many harmonics would be extremely difficult, because
it would require a very fine, regular and extensive sampling of CFs.
We overcame this
hurdle by instead applying the principle of local "scaling invariance" in cochlear mechanics
(Zweig 1976). Scaling invariance means that, at any particular location xo on the basilar
membrane whose CF is CFo, the response H (magnitude and phase) of the basilar membrane
to a given frequencyf can be expressed as a function only of the ratio
CF
That is to say,
H(f,xo) = H( CF). Therefore,for a stimulusof frequencyfl, H(f1 ,x o) = H( CF). If
we call fi the ratio between f and f (
f
= ), then H(fl, xO)= H(
CF)
=
1.
The magnitude and phase of the response measured at xo to a frequency fj are therefore equal
to the magnitude and the phase of the response to f at the cochlear location tuned to CF
.
This means that the waveforms of the two responses have the same shape and they are scaled
in time by a factor f.
By varying fl (and therefore fl) and measuring responses at a fixed
location tuned to CFo, one can therefore infer the response to a stimulus of fixed frequencyf
at different cochlear locations.
Assuming that this scaling property holds for each of the frequency components of a
complex sound, and that the locality of this principle is not violated (as would be the case if
we were interested in single-fiber responses to frequencies very far away from CF), the
spatio-temporal
response pattern to a complex tone with a given FO is equivalent to the
response at a given CF to a complex tone with varying FO, if cochlear place and time are
plotted in dimensionless units CF/FO ("harmonic number") and txFO ("normalized time"),
respectively.
This principle is very effectively visualized in Figure 2.2.
56
On the left is the spatio-temporal response pattern predicted by the Zhang et al. model,
for AN fibers with CFs spanning the range between 750 and 2250 Hz, to a complex tone with
fixed F0 at 500 Hz. The harmonic number varies from 1.5 to 4.5.
On the right is the response pattern predicted by the same model, for a single AN fiber
with CF at 1500 Hz, to complex tones whose Fs
encompass the range between 333 and
1000 Hz, such that the harmonic number varies over the same range (1.5 to 4.5) as in the
figure on the left. Here, the time axis is expressed in dimensionless units T = txF0 (number
of cycles).
The response patterns for the two conditions (the stimulus F0 is fixed on the left, while
the CF is fixed on the right) are nearly undistinguishable:
changes in correspondence
they both show fast latency
of integer values of harmonic number, and relatively more
constant latency for harmonic numbers that are integer multiples of 0.5 (2.5, 3.5, etc.).
Average discharge rate and mean absolute spatial derivative, computed as for Fig. 2.1, also
exhibit nearly identical features (peaks at integer values of harmonic number and valleys in
between).
Thus, the results of these model simulations support the effectiveness of our strategy of
relying on scaling invariance to infer the response of a population of AN fibers to a complex
tone of a given F0 by measuring the response of a single AN fiber to complex tones with
varying FO.
In this study, we tested the spatio-temporal representation of the pitch of complex tones
by recording from single AN fibers in anesthetized cats. We found that this representation is
viable for F
in the range of cat vocalizations,
and that it may overcome some of the
limitations of the rate-place and of the interval representation
in accounting for trends
observed in human psychophysics experiments. Preliminary reports of our findings have
been presented (Cedolin and Delgutte 2004, 2005a).
57
2.2 MATERIALS AND METHODS
Procedure
Methods for recording from auditory-nerve fibers in anesthetized cats were as described
by Cedolin and Delgutte (2005b). Cats were anesthetized with Dial in urethane (75 mg/kg),
with supplementary doses given as needed to maintain an areflexic state.
The posterior
portion of the skull was removed and the cerebellum retracted to expose the auditory nerve.
The tympanic bullae and the middle-ear cavities were opened to expose the round window.
Throughout the experiment the cat was given injections of dexamethasone (0.26 mg/kg) to
prevent brain swelling, and Ringer's solution (50 ml/day) to prevent dehydration.
The cat was placed on a vibration-isolated table in an electrically-shielded,
temperature-
controlled, sound-proof chamber. A silver electrode was positioned at the round window to
record the compound action potential (CAP) in response to click stimuli, in order to assess
the condition and stability of cochlear function.
Sound was delivered to the cat's ear through a closed acoustic assembly driven by an
electrodynamic
speaker (Realistic 40-1377). The acoustic system was calibrated to allow
accurate control over the sound-pressure
level at the tympanic membrane.
Stimuli were
generated by a 16-bit digital-to-analog converter (NI 6052E) using sampling rates of 20 kHz
or 50 kHz. Stimuli were digitally filtered to compensate for the transfer characteristics of the
acoustic system.
Spikes were recorded with glass micropipettes filled with 2 M KC1. The electrode was
inserted into the nerve and then mechanically advanced using a micropositioner (Kopf 650).
The electrode signal was bandpass filtered and fed to a custom spike detector. The times of
spike peaks were recorded with -,us resolution and saved to disk for subsequent analysis.
A click stimulus at approximately 55 dB SPL was used to search for single units. Upon
contact with a fiber, a frequency tuning curve was measured by an automatic tracking
algorithm (Kiang et al. 1970) using 50-ms tone bursts, presented at a rate of 10/s, and the
characteristic frequency (CF) was determined. The spontaneous firing rate (SR) of the fiber
was measured over an interval of 20 s. The responses to complex-tone stimuli were then
studied.
58
Stimuli
Stimuli were harmonic complex tones with missing fundamentals. The fundamental
frequency (F0) of a complex tone was stepped up and down over a range such that the ratio
of fiber's CF to F0 ("harmonic number") typically varied from 1.5 to 4.5, in order to capture
low-order harmonics, likely to be resolved. For each fiber, the values of F0 were chosen so
as to regularly sample the range of harmonic numbers used. The typical harmonic-number
step was 1/8. Therefore, the typical number of F0 for which responses were collected in each
fiber was 25 (8 x (4.5 - 1.5) + 1). Each complex tone was composed of harmonics 2 to 20,
all of equal amplitude, and the fundamental frequency was always missing. Each of the F0
steps lasted 200 ms, including a 20-ms transition period during which the waveform for one
F0 gradually decayed while overlapping with the gradual build up of the waveform for the
subsequent F0. Responses were typically collected over 20 repetitions of the stimulus, with
no interruption.
The sound pressure level of each harmonic was initially set at 10-15 dB above the fiber's
threshold for a pure tone at CF. When possible, the stimulus level was then varied over a
wide range (typically 20-30 dB) to investigate the stability of the spatio-temporal
representation at levels at which the average-rate response is saturated.
In order to compare neural responses to psychophysical data on the phase dependence of
pitch, three versions of each stimulus were generated with different phase relationships
among the harmonics: cosine phase, alternating (sine-cosine) phase, and negative Schroeder
phase
(Schroeder
1970).
The three
stimuli
have
the same
power
spectrum
and
autocorrelation function, but differ in their temporal envelope (Figure 2.3). The cosine-phase
stimulus has a very "peaky"
alternating-phase
envelope with periodicity
at F.
The envelope of the
stimulus is also very peaky, but its periodicity is at 2xF0, even though the
period of the waveform remains equal to /F0. Finally, a Schroeder phase relationship
among the harmonics produces a waveform whose envelope has minimum oscillations.
While markedly dissimilar in their envelope, stimuli with harmonics in these three phase
configurations produce similar pitches and pitch strengths in psychophysical experiments, as
long as some o the harmonics are resolved (Houtsma and Smurzynski 1990; Carlyon and
Shackleton 1994).
59
Analysis
In response to each complex tone, we constructed
period histograms
using spikes
occurring in a 180-ms window extending over each F step, but excluding the transition
period between F
steps.
The non-scaling conduction
delay typical for each cochlear
location (Carney and Yin 1988) was subtracted from the timing of each spike.
Period
histograms were visualized as a function of both time and F0, as in Fig. 2.2B. Consistent
with scaling invariance, the time and F0 axes were expressed in dimensionless units T = txF0
(number of cycles) and n = CF/F0 (harmonic number, or normalized place), respectively.
As
a consequence of this normalization of the time axis, the number of bins over which the
period histograms were computed was the same (50) for all FOs.
Two metrics were
computed from the resulting spatio-temporal response pattern H(n,7T):the average discharge
rate ("Rag"), obtained by integrating each period histogram over time and converting to
spikes/s, and the "mean absolute spatial derivative" ("MASD"), obtained by differentiating
the time-F0 pattern with respect to harmonic number, then taking the absolute value and
integrating over time.
Ravg(n)= fH(n,T)dT;
MASD(n) =
an
dT
0
The spatial derivative represents the effect of a hypothetical central mechanism based on
lateral inhibition that could, in principle, extract spatio-temporal pitch cues (Shamma 1985b).
Both Ravgand MASD were smoothed by linear convolution with a three-point triangular
filter and then plotted as functions of harmonic number. Integer values of harmonic number
occur when the F
is such that a fiber's CF coincides with one of the harmonics of the
stimulus, while the harmonic number is an odd integer multiple of 0.5 (2.5, 3.5, etc...) when
the F0 is such that the CF falls halfway between two harmonics.
Resolved harmonics are
expected to result in peaks in either the Ravg or the MASD (or both) for integer values of the
harmonic number, with valleys in between.
As in our previous study (Cedolin and Delgutte 2005c), we used "bootstrap" resampling
(Efron and Tibshirani 1993) of the data recorded from each fiber, in order to evaluate the
statistical properties
of the estimates of Ravg and MASD.
In particular, one hundred
resampled data sets were generated by drawing with replacement from the set of spike trains,
60
typically recorded over 20 stimulus repetitions. Response intervals corresponding to the
ascending and descending part of the F sequence were drawn independently from each
other. Average discharge rate and mean absolute spatial derivative were computed from each
bootstrap data set, and the standard deviations of these measures were used as error bars.
A simple mathematical function was fit to a fiber's response to the complex-tone stimuli.
The function, which can capture the main features of both the Ravg and the MASD, is the
following:
-n
f(n)=A.cos(2r.
.n)
en
-n
+B e n +C
where n is the harmonic number (CF/F0).
(1)
A co-sinusoidal oscillating component at
frequency , whose amplitude decays exponentially with harmonic number n (CF/F0), is
added to a constant term C and to an exponential function of harmonic number. In the case
where 5 is equal to one, the function peaks exactly at integer values of harmonic number,
effectively resulting in the "characteristic frequency" of the oscillating component (the
product of c and the pure-tone CF) to match the tuning-curve CF.
The model has 5 free parameters: the amplitude (A) and frequency () of the oscillating
component; the time constant (no) of the decay of the oscillation (fixed to be equal to the time
constant of the baseline trend); the amplitude (B) of the baseline increase (or decrease); the
value of the DC component (C). The model was fit independently to profiles of Ravg and
MASD
by the least squares
method
using
the Levenberg-Marquardt
algorithm
as
implemented by Matlab's "lsqcurvefit" function. The procedure was repeated for each of the
one hundred profiles of Ravg and MASD derived from the bootstrap data sets to quantify the
variability of parameter estimates.
Based on the values of the parameters of the best fitting curves, we defined metrics that
were used to directly compare precision, accuracy and strength of the pitch cues provided by
the rate-place and the spatio-temporal representation.
61
2.3 RESULTS
Our results are based on 173 measurements of responses to harmonic complex tones
recorded from 94 auditory-nerve fibers in 6 cats. Of these, 76 had high SR (> 18 spikes/s), 3
had low SR (< 0.5 spike/s), and 15 had medium SR. The CFs of the fibers ranged from 300
Hz to 5 kHz.
Figure 2.4 shows the responses to complex tones with harmonics in cosine phase for
three auditory-nerve
respectively.
fibers with CFs of 700 Hz (A), 2150 Hz (B) and 4300 Hz (C),
The responses shown were measured at 10, 20 and 30 dB, respectively, above
each fiber's threshold for a pure-tone at CF. Period histograms are displayed as a function of
both normalized time (txF0, horizontal axis) and harmonic number (CF/F0, vertical axis) (see
Methods) in the left panels. The stimulus waveform is also plotted against normalized time.
The right panels show the average discharge rate
(Ravg)
and the mean absolute spatial
derivative (MASD) as a function of harmonic number derived from the corresponding spatiotemporal response pattern (see Methods).
For the low-CF fiber, the latency of the response varies more or less uniformly with
harmonic number, even at very low (10 dB) level above threshold.
Strong cues to resolved
harmonics are therefore not present in the spatio-temporal response pattern of this fiber. This
is reflected in the absence of pronounced peaks in the MASD at integer harmonic numbers.
The same observation is valid for the
Ravg,
which is nearly constant.
This suggests that, at
this CF, both the rate-place and the spatio-temporal representations do not provide strong
evidence for resolved harmonics for the F0s in the range used.
The spatio-temporal pattern of the response of the fiber with CF at 2150 Hz does show
variations in response latency with harmonic number, qualitatively similar to those predicted
by the spatio-temporal model of Fig. 2.2.
The phase varies rapidly at integer harmonic
numbers, while it changes more slowly at harmonic numbers that are odd integer multiples of
0.5. As a result, the MASD shows local maxima at harmonic numbers equal to 2, 3 and 4,
and local minima in between, thus providing evidence for spatio-temporal cues to pitch in the
AN response.
The
Ravg
also shows peaks at integer harmonic numbers, even though they
appear less pronounced than those of the MASD.
62
As CF increases, the ability of single AN fibers to resolve harmonics of complex stimuli
based on average rate is expected to improve, due to the progressive sharpening of the
relative bandwidth of cochlear filters with respect to their center frequency (Kiang et al.
1965; Shera et al. 2002; Present study, Fig. 1.3 and 1.4). On the other hand, the effectiveness
of the spatio-temporal representation may decrease at high CFs, due to the rapid worsening in
phase-locking
above 2-3 kHz (Johnson 1980) in the cat AN.
confirmed for a fiber with a CF of 4.3 kHz, in Fig. 2.4C.
threshold), up to 6 harmonics are resolved based on
Ravg,
Both these predictions are
At this level (30 dB above
while no strong latency cues are
apparent from examining the spatio-temporal response pattern. Although small peaks in the
MASD are present at the second and third harmonic number, they appear to reflect
predominantly the variation in average rate with harmonic number, rather than differences in
response latency.
As pointed out in the introduction, an appealing aspect of the spatio-temporal
representation
of pitch is that it is expected to work at levels at which the rate-place
representation breaks down. To test this hypothesis, we compared the effectiveness of the
two representations at different stimulus levels. Figure 2.5 shows data for one AN fiber with
pure-tone CF at around 1920 Hz. Responses to complex tones with harmonics in cosine
phase were recorded at 10, 25 and 40 dB, respectively, above the fiber's threshold at CF (25
dB SPL). At the low level (top), the second, the third, and arguably the fourth harmonic
appear as distinct peaks in both the Ravg and the MASD. As the level of each harmonic is
increased to 25 dB above threshold (middle), the Ravgbegins to show signs of saturation as
only the second harmonic appears to be resolved.
In contrast, strong latency cues to the
second, third and arguably fourth harmonic are still present in the spatio-temporal response
pattern, resulting in corresponding prominent peaks in the MASD. At the highest level tested
(65 dB SPL, bottom) the
Ravg
is completely saturated, while peaks at the second and third
harmonic number are still easily detectable in the MASD. This example seems therefore to
support our hypothesis that a spatio-temporal representation might still work at levels at
which the effectiveness of a strictly rate-based representation is decreased.
63
Populationresults
To quantitatively compare the precision and the strength of the pitch cues provided by the
rate-place and by the spatio-temporal representation, a mathematical function (see Methods)
was fit independently to profiles of Ravg and MASD for each fiber in our sample.
An
example is shown in Figure 2.6, for the same fiber as in Fig. 2.4B. The tuning-curve CF of
this fiber was 2150 Hz and the level of each of the cosine-phase harmonics was 20 dB above
the threshold at CF. The best-fitting curves (solid lines) closely capture the oscillations of
The "characteristic
both the Ravg and the MASD.
frequency" CFosc of the oscillating
component of the fitted curve (the product of the tuning-curve CF and the parameter 5 of the
best-fitting curve) is effectively an estimate of the characteristic frequency of the fiber based
on its response to complex tones.
When
is equal to one, the function peaks exactly at
integer values of harmonic number, effectively resulting in CFoscto match the tuning-curve
CF.
In the example of Fig. 2.6, the estimate of the CF provided by the MASD fit is 2263 ±+4
Hz, while the one based on the
Ravg
fit is 2210 ± 5 Hz. Both are slightly larger than the
estimate based on the pure-tone tuning curve (2150 Hz).
Figure 2.7 shows the relationships between the estimates of CF derived from profiles of
Ravg
and MASD and from pure-tone tuning curves (TCs) for our entire sample of fibers.
Differences between CF estimates, expressed as a percentage of the tuning-curve CF, are
plotted against tuning-curve CF. Data are grouped by level relative to each fiber's threshold
for a pure tone (columns).
For a point to be included in this figure (and in all the following
summary plots), the variance of the residuals after fitting the single-fiber model to the data
had to be significantly smaller (P < 0.05, F-test) than the variance of the residuals for a model
without the oscillating component (see Fig. 2.6), so that the parameter 6 (and therefore the
CF) could be reliably estimated. Excluded measurements are indicated by black crosses in
Fig. 2.7.
Analysis of covariance was performed on the data displayed in each row of Fig. 2.7 to
quantify whether the data depended on CF, stimulus level, or a combination of both. Red
lines indicate the linear models with the minimum number of parameters (based on F-test on
the variance of the residuals) that provided the best fits (on a log-frequency scale) across all
64
level groups.
Results of this analysis are stated only if statistically significant at the 0.01
level (or better).
At low CFs, differences between MASD-based and TC-based CF estimates are typically
positive and can be as large as 20-25% (Fig. 2.7, A-C). These deviations exhibit a similar
decrease with CF in all three level ranges (red lines).
The Ravg-based estimates of CF are
also typically larger than the TC-based ones at low CFs (Fig. 2.7, D-F), and these differences
also decrease significantly with CF at all levels (red lines). The best-fitting linear model has
fixed slope (-13.92) across levels, but its intercept decreases significantly at the highest
levels. For fibers with CFs above about 3 kHz, tuning curve CFs tend to become larger, on
average, than Ravg-based CFs at very high levels (F).
MASD-based
(Fig. 2.7, G-I).
CF-estimates are generally larger than Ravg-based estimates at all levels
The trend in the data showed no significant dependence
on CF, but a
significant effect of level (the mean difference going from 0.75% at low levels (G) up to
2.67% at the highest levels (I)). This result suggests that, at a particular cochlear place, the
pure-tone frequency for which the magnitude of the basilar membrane response is maximum
is lower than the frequency for which the rate of variation of the phase of the response is
fastest. This is true even at very low levels, and the effect grows with stimulus level.
By virtue of the scaling invariance principle, and for both the Ravg and the MASD, the
statistical properties of an estimate of a fiber's CF provided by the single-fiber model should
be analogous to those of a hypothetical estimate of the pitch of the complex tone that would
produce a spatio-temporal response pattern equivalent to the one observed (Fig. 2.2). In
particular, precision and strength of an estimate of CF yielded by the single-fiber model can
be considered as equivalent to the precision and the strength of pitch estimation, if the
appropriate conversion from CF back to F0 is made (we will return to this point with more
detail in the Discussion).
A
measure of precision of the CF estimates obtained from best-fitting single-fiber models
is shown in Figure 2.8 for profiles of
Ravg
(black) and MASD (red). The standard deviation
of the estimates across bootstrap trials, expressed as a percentage of the tuning-curve CF, is
plotted against CF for low (A), moderate (B) and high (C) stimulus levels relative to each
fiber's threshold at CF.
With few exceptions, estimates are highly precise for both
representations, up to the highest levels tested, as their standard deviations are in most cases
65
within 2% of the tuning-curve CF. At about 3 kHz and above, the few reliable estimates of
CF derived from model MASD appear to be more variable than those derived from model
Ravg,
suggesting that the degradation of phase-locking at higher frequencies limits latency
cues and thus impairs the effectiveness of the spatio-temporal representation.
In contrast, the
rate representation continues to improve due to better harmonic resolvability.
The more pronounced the oscillations in the Ravgand in the MASD, the better individual
harmonics are considered to be resolved.
To quantify this observation, we use the area
between the top and bottom envelope of the oscillating component of the fitted curve (light
shadings in the examples of Fig. 2.6) as a measure of the strength of average-rate and spatio-
temporal representations. Since the area within the oscillation is very different for profiles of
Ravg
compared to profiles of MASD, we use the ratio of the area within the oscillation to the
typical standard deviation of the data points (dark shadings in Fig. 2.6) to directly compare
the strengths of the two representations.
We call this ratio the "harmonic strength" of the
MASD and of the Ravg, respectively.
Harmonic strengths, computed for both the Ravg (black) and the MASD (red), are plotted
against CF in Figure 2.9 for our entire sample of responses to complex tones with harmonics
in cosine-phase. Below about
kHz, where individual harmonics are poorly resolved, the
harmonic strengths of both Ravg and MASD are low at all levels. For CFs above 1 kHz, the
harmonic strengths for both representations are highest at low levels (A), while they tend to
progressively decrease at higher levels (B-C). This decrease may be due to broadening of
peripheral filters, rate saturation (in the case of the
about 3 kHz, the harmonic strengths of the
Ravg
Ravg),
or both. However, for CFs above
at the highest levels tested (C) often remain
as large as at low and medium levels. For CFs above 3 kHz, the fraction of measurements
for which a curve with an oscillating component gave only an insignificant improvement
over a curve without an oscillation when fitting profiles of MASD, increased dramatically
with level (from 8/26 at low and medium levels up to 10/16 at high levels), indicating that the
spatio-temporal representation degrades rapidly with level in this frequency range.
To directly compare the strengths of the rate-based and of the spatio-temporal
representations, we defined a metric that we call "normalized strength difference" as the
difference between the harmonic strength of the MASD and that of the Ravg, divided by their
sum. For example, if the harmonic strength of the MASD is three times the harmonic
66
strength of the Ravg, the corresponding normalized strength difference is equal to 0.5, while if
the harmonic strength of the MASD is one-third of the harmonic strength of the Ravg, the
normalized strength difference is -0.5. This metric assumes values between - I and 1, and it
is chosen because reciprocal ratios of harmonic strengths result in normalized strength
differences with the same absolute value but opposite sign.
Normalized strength differences are shown against CF in Figure 2.10 for those
measurements for which profiles of MASD and
curve including an oscillating component.
Ravg
oscillated sufficiently to reliably fit a
Two main trends are noticeable, for CFs below
and above about 3 kHz, respectively.
For CFs between 1 and 3 kHz, the strengths of the two representations
are generally
comparable at low levels (A). In this range of CFs, as level increases the MASD tends to
produce larger harmonic strength than the
Ravg,
resulting in normalized strength differences
that are more often positive than negative at moderate levels (B). At high levels (C), the vast
majority of normalized strength differences are positive, indicating that although both
representations are less effective than they are at low levels (Fig. 2.9), the oscillations in the
spatial derivative remain relatively more pronounced than those in the average rate.
An
analysis of covariance for fibers with CFs between 1 and 3 kHz (red lines) showed a decrease
of normalized strength differences with CF at all levels (with same slope) and a significant
increase of the intercept of the best-fitting line with level.
For CFs above about 3 kHz, normalized strength differences
tend to assume large
negative values at all levels, indicating that rate cues to resolved harmonics are significantly
stronger than latency cues. An analysis of covariance for fibers with CF between 3 and 5
kHz (purple lines) showed a significant decrease of normalized strength differences with CF
and no significant effect of level.
This result is most likely caused by the worsening of
phase-locking with the increasing frequency of the (resolved) harmonics to which high-CF
fibers are tuned. This phenomenon has already been pointed out as a possible cause for the
higher variability of MASD-based CF estimates at high frequencies relative to that of
based estimates (Fig. 2.9).
67
Ravg -
Phase effects
The results of psychophysical studies with complex tones containing resolved harmonics
(Houtsma and Smurzynski 1990; Shackleton and Carlyon 1994) are generally not greatly
affected by the relative phase relationship among the partials. To test whether the rate-place
and spatio-temporal representations are consistent with this finding, we also measured
responses to complex tones with harmonics in alternating phase and in Schroeder phase.
Figure 2.11 shows an example for a fiber with a CF of 2520 Hz. The lowest F0 presented in
our stimulus was therefore 560 Hz.
In general, for fibers with CF around 2500 Hz, the
lowest F0 for which harmonics are resolved in the average-rate response should be around
400-500 Hz (Present study, Fig. 1.4). Thus, we can assume that in our example, the stimulus
contained well-resolved harmonics at all FOs. The spatio-temporal response patterns are very
similar across phase conditions (Fig. 2.11, A-C), as are the strong pitch features present in
both
Ravg
and MASD (Fig. 2.11, D,E).
A comparison of the strengths of the cues to resolved harmonics provided by the two
representations in the three phase conditions is shown in Figure 2.12.
The CFs of the AN
fibers included in these plots varied from just above 1 kHz to 3 kHz. Harmonic strengths of
the MASD for harmonics in cosine and alternating phase are very similar (A), and both are
only slightly larger (non-significantly at the 0.01 level, paired t-test) than the harmonic
strengths for harmonics in Schroeder phase (B-C). Similarly, we found no statisticallysignificant difference for harmonic strengths of the Ravg for harmonics in the three phase
configurations (D-F), in agreement with our previous findings (Cedolin and Delgutte 2005c).
These results suggest that the salience of pitch cues available in both the response latency
and magnitude of single AN fibers is to a large extent independent of the phase relationship
among resolved harmonics, consistent with observations from studies of human
psychophysics.
2.4 DISCUSSION
We investigated the effectiveness of a spatio-temporal representation of the pitch of
complex tones in the cat auditory nerve. This study was based upon the hypothesis that cues
68
to resolved harmonics might be generated by rapid changes in the latency of the basilar
membrane response at cochlear locations tuned to each resolved harmonic frequency (Fig.
2.1).
This code is "spatio-temporal"
in that it requires a combination of resolvability of
individual harmonics and effective phase-locking at their frequencies.
The most direct method to appropriately test this hypothesis would be to measure the
response of an array of AN fibers with very finely and regularly spaced characteristic
frequencies, spanning a range wide enough to encompass several harmonic frequencies.
Because of the excessive difficulty in making this measurement, we used a simplified, yet
nearly equivalent approach based on the principle of scaling invariance in cochlear
mechanics (Zweig 1976). The scaling invariance principle allows us to infer the spatio-
temporal response pattern of the AN to a complex tone with a fixed FO from the response
pattern of a single AN fiber, measured as a function of the FO of a complex tone (Fig. 2.2).
Profiles of "mean absolute spatial derivative" (MASD) and "absolute discharge rate"
(Ravg) were derived from single-fiber spatio-temporal discharge patterns.
The MASD
operates on a spatio-temporal discharge pattern to simulate the extraction of spatio-temporal
cues to resolved harmonics by a putative lateral-inhibition
mechanism, sensitive to the
timings of its inputs, which are assumed to originate from neighboring cochlear locations.
The Ravg is used to quantify
the cues to pitch provided by a traditional
rate-based
representation. Simple mathematical functions were fit to profiles of both MASD and
(Fig. 2.6), and the parameters
of the best-fitting
curves were used to compare
Ravg
the
effectiveness and strength of the two representations.
Whenever possible, we varied the stimulus level over a wide range to test whether the
spatio-temporal representation would still hold at levels at which the rate-based
representation is no longer viable.
Spatio-temporalcues to resolvedharmonicsin singleANfiber
responses
Our results show that strong spatio-temporal cues to resolved harmonics are available in
the response of AN fibers whose CFs are high enough for harmonics to be sufficiently
resolved (above around 1 kHz) but below the limit (around 3 kHz) above which phase-
locking is significantly degraded (Fig. 2.4). For fibers with CF below
69
kHz, rate-based cues
to resolved harmonics are also generally weak, due to poor harmonic resolvability. At high
CFs, on the other hand, the strength of the rate-place representation
grows, due to the
progressive sharpening of cochlear filters with their characteristic frequency (Kiang et al.
1965; Shera et al. 2002,
Cedolin
and Delgutte
2005c),
while the spatio-temporal
representation breaks down due to the rapid degradation of phase-locking with frequency
(Johnson, 1980).
Estimates of CF based on profiles of MASD were generally slightly higher than those
based on Ravg (Fig. 2.7, G-I). The differences between the two increased significantly with
level.
We will briefly discuss the implications of this finding in the next section.
Both
MASD- and Ravg-basedestimates of CF were generally higher than CFs estimated from the
pure-tone tuning curves (Fig. 2.7, A-F) at low levels, and this effect was larger at low CFs.
One possibility is that the particular method (Kiang et al. 1970) we used to measure pure-
tone tuning curves could have led to estimates of CF with a slightly negative bias.
Thresholds were measured for tone frequencies in decreasing order, and each threshold was
used as a starting level for the threshold-seeking
frequency.
algorithm at the next (slightly lower)
For tone frequencies on the high-frequency side of the true CF, each starting
condition is therefore likely to be a frequency-level combination falling in a highly excitatory
area, while the opposite is true for tone frequencies lower than the true CF.
Due to the
stochastic nature of AN fibers responses, there is a bias towards overestimating thresholds on
the high-frequency
side of the true-CF and thus to underestimating
the value of the CF.
However, it is not clear whether this effect should be more pronounced for high-CF fibers
because their tuning curves are steeper, on the high-frequency side, than for low-CF fibers, or
vice versa, as we observed.
When the MASD- and Ravg-basedestimates of CF were reliable, their standard deviations
were generally within a few percent at all levels (Fig. 2.8), with larger deviations becoming
more common for the MASD-based estimates at CFs above about 3 kHz.
For CFs between 1 and 3 kHz, the strengths of the spatio-temporal representation and of
the rate-place representation were comparable at low levels (Fig. 2.9A, 10A), but as level
increased the spatio-temporal representation was increasingly more effective than the rate
representation (Fig. 2.10B,C). The rate-place representation degraded very rapidly with
level, due to broadening of cochlear filters (Ruggero et al. 1992) and saturation of single AN
70
fibers responses (Cedolin and Delgutte 2005c). The spatio-temporal representation also
shows signs of deterioration with level (Fig. 2.5), consistent with the finding that responses
from fibers innervating neighboring cochlear locations become more coincident at high
levels (Anderson 1971; van der Heijden and Joris 2003, Carney and Yin 1988).
The profiles of MASD and Ravg derived from spatio-temporal patterns of response to
stimuli with harmonics in cosine, alternating and Schroeder phase were very similar for CFs
in the range where the harmonics of the stimuli were likely to be resolved (Fig. 2.11). In this
CF-range, the strength of both representations also did not vary significantly with phase (Fig.
2.12). Together, these findings suggest that the spatio-temporal representation produces
results that are broadly consistent with the lack of phase effects observed in psychophysics
experiments when the stimuli contain resolved harmonics.
Scaling invariance "unfolded"
So far, we have discussed the results of our experiment in terms of the characteristic
frequencies of the single AN fibers whose spatio-temporal response patterns were measured
as a function of complex-tone
F.
However, it is interesting
to "unfold" the scaling
invariance principle and try to draw conclusions in terms of stimulus F0. Since, for the vast
majority of our stimuli, we used FOssuch that the harmonic number (the ratio of fiber's CF to
stimulus F0) varied from 1.5 to 4.5, on the average the single-fiber spatio-temporal discharge
pattern observed at a given CF most closely corresponds to the one that would be generated
by a single complex tone with F0 roughly equal to CF/3 (Fig. 2.2). This conversion from CF
to F
is also supported by the observation
that an unambiguous determination
of the
(missing-fundamental) stimulus F0 can only be possible if at least 2 of the harmonics are
resolved. All our results can therefore be reinterpreted in terms of F0 by simply dividing the
CF at which the recording was made by three.
As we have seen, the spatio-temporal representation works best for CFs in the range
roughly between I and 3 kHz. This range would correspond to a range of F0 between 333
Hz and 1 kHz, covering the entire range of conspecific vocalizations in the cat.
For FOs between 333 Hz and 1 kHz, the rate-place and the spatio-temporal representation
are equally effective at low levels (Fig. 2.10OA). As stimulus level increases, although both
representations degrade, the strength of the spatio-temporal representation exceeds that of the
71
rate representation (Fig. 2.10 B,C).
For Fs
below 333 Hz, both the rate-place and the
spatio-temporal representation work poorly at all levels, indicating insufficient harmonic
resolvability in the cat. This limit is broadly consistent with the finding of our previous study
(Cedolin and Delgutte 2005c) that pitch estimation from rate-place profiles was problematic
for F0s below 400-500 Hz (although some reliable estimates could be obtained for F0s as
low as 250 Hz). For F0s above 1 kHz, as harmonics become increasingly resolved due to the
progressive sharpening of cochlear filters relative to their center frequency, the rate
representation becomes increasingly more effective (Fig. 2.9). Conversely, for missingfundamental stimuli, phase-locking at the frequency of the harmonics worsens dramatically
for Fs
above 1 kHz.
This results in the spatio-temporal representation becoming
progressively weaker (Fig. 2.10) and less precise (Fig. 2.8) than the rate-representation.
It is interesting to attempt an extension of these cat findings to humans. Two important
assumptions have to be made. First, that the lower limit for the viability of a spatio-temporal
representation (around 333 Hz in cats) is determined by sharpness of tuning. Second, that
phase-locking is primarily responsible for the decreased effectiveness of a spatio-temporal
representation above an upper F0-limit (of about 1 kHz in cats). Although the result is a
matter of current debate (Ruggero 2005), it has been argued (Shera et. al 2002, Oxenham and
Shera, 2003) that the frequency tuning of the auditory periphery in humans might be up to 3
times shaper than in cats. There are no strong reasons to assume that the degree of neural
phase-locking and its upper limit should be significantly different for humans than for cats.
Based on these assumptions, the lower F0-limit for a spatio-temporal representation of the
pitch of complex tones with a missing fundamental might translate from about 333 Hz in cats
to about 111 Hz in humans, while the upper limit should remain the same, at about
kHz.
This range encompasses most of the range of F0 of human voice (80-300 Hz), and the upper
limit is consistent with the existence of an upper limit to the pitch of missing-fundamental
stimuli [about 1400 Hz in humans (Moore 1973b)].
Another interesting result of our study was that the estimates of CF derived from profiles
of spatial derivative were slightly, but systematically larger than those derived from profiles
of average rate (Fig. 2.7, G-I), even at low stimulus levels. As stated in the results section,
this suggests that, at a particular cochlear place, the pure-tone frequency for which the
magnitude of the basilar membrane response is maximum is lower than the frequency for
72
which the rate of variation of the phase of the response is fastest. The magnitude of this
effect increases significantly with stimulus level. Unfolding scaling invariance, these
differences indicate that, for a pure-tone stimulus, the magnitude of the basilar membrane
response might be maximum at a cochlear place increasingly basal with level to the cochlear
place at which the phase varies most rapidly. Although, to our knowledge, this effect has not
been systematically investigated for a wide range of CFs in studies of basilar membrane
mechanics, there is evidence for a downwards shift of BF with level at frequencies above 1
kHz in the basilar membrane response of the guinea pig (de Boer and Nuttal 2000) and in the
AN response of the cat (Greenberg et al. 1986) and of the squirrel monkey (Rose et al. 1971).
A study by Shera (2001) suggests that these shifts with level do not originate because of local
changes of the resonant frequency of the cochlear partition, but are consequences of the
global increase of the driving pressure with stimulus level.
Possiblemechanismsfor the extractionof spatio-temporalpitch cues
So far we have discussed the availability of cues to resolved harmonics of complex tones
in the spatio-temporal pattern of activity of tonotopically-arranged neurons at the level of the
auditory nerve. These spatio-temporal cues to harmonic frequencies could theoretically be
extracted by neurons that are (1) innervated by auditory nerve fibers with neighboring
characteristic frequencies and (2) sensitive to slight differences in the timing of their inputs.
One possible mechanism is lateral inhibition, likely producing local maxima in discharge
rate as inputs with neighboring CFs become less coincident (as would be the case when the
phase change with CF is fast). An alternative decoding mechanism is cross-frequency
coincidence detection (Carney 1994; Heinz et al. 2001) which would result in local minima
at locations corresponding to the harmonics since those are the places where the rapid phase
changes would cause inputs with neighboring CFs to be less coincident.
Carney (1990) showed that some neurons in the posterior antero-ventral cochlear nucleus
(AVCN), most frequently with primarylike-with-notch, transient-chopper and onset response
characteristics, are sensitive to the relative timing of their inputs. Precisely, the discharge
rate of some of the cells in response to Huffman sequences (clicks filtered in such a way that
their magnitude spectra remain flat while a phase variations of desired width is introduced at
the neuron's CF) varied with the width of the phase transition. Although many cells
73
responded more vigorously to shallow phase transitions, indicating "preference" for more
coincident inputs, suggesting coincidence detection as the mechanism extracting phase
information, in some cases this trend was reversed, consistent with lateral inhibition.
These
finding suggest that coincidence detection and lateral inhibition might not be mutually
exclusive in the AVCN.
Interestingly, for some cells the preferred slope of the phase
transition depended on stimulus level (typically being shallow at low levels and steep at high
levels), indicating that coincidence detection and lateral inhibition might actually be
implemented in the same cells at different stimulus levels.
2.5 CONCLUSIONS
We tested the effectiveness of a spatio-temporal representation of the pitch of complex
tones in the cat auditory nerve at a wide range of stimulus level.
We found that spatio-
temporal cues to resolved harmonics are available in the cat AN for FOs in the range between
333 Hz and 1 kHz and can, in principle, be extracted by a lateral-inhibition mechanism.
We
argue that the lower FO-limit for the spatio-temporal representation is primarily determined
by the limited frequency selectivity of auditory-nerve fibers at low CFs, while the upper limit
is caused by the abrupt degradation of phase-locking for AN fibers with CFs above 3 kHz.
The spatio-temporal representation is viable in the entire FO-range of cat vocalizations, and
the same result may be extended to humans by taking into account their sharper cochlear
frequency tuning. The spatio-temporal representation is consistent with the existence of an
upper FO-limit to the perception of the pitch of complex tones with a missing FO, and its
strength does not depend on the relative phase between the harmonics, when these are
resolved. The spatio-temporal representation is thus consistent with trends observed in many
studies of human psychophysics,
and it is more effective than the classical rate-place
representation at high stimulus level.
74
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complex tone with FO of 200 Hz at 50 dB SPL. The response is displayed as a function of
time and harmonic number (CF/FO). Fast variations in response latency with CF at integer
harmonic numbers are highlighted in yellow. The resulting profiles of average rate (black)
and mean absolute spatial derivative (red), normalized by their maximum values, are shown
on the right.
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the Zhang model for AN fibers with CFs from 750 to 2250 Hz to a complex tone with FO of
500 Hz at 40 dB SPL, displayed as a function of harmonic number
response pattern generated using the same model for one AN
complex
(CF/FO) and time. B:
fiber with CF of 1500 Hz to
tones at 40 dB SPL with FOs from 333 to 1000 Hz, displayed as a function of
harmonic number
and normalized time txFO. Solid lines show the profiles of normalized
average discharge rate (black) and mean absolute spatial derivative (red) derived from the
two response patterns.
76
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Figure 2.3: different phase relationships among the harmonics give rise to different stimulus
waveforms. For harmonics in cosine phase (top), the waveform shows one peak per period of
the FO. When the harmonics are in alternating phase (middle), the waveform peaks twice
every period of the FO. A negative Schroeder phase relationship among the harmonics
(bottom) minimizes the amplitude of the oscillations of the envelope of the waveform.
77
Figure 2.4. (Next page) Response patterns of three AN fibers with CFs of 700 Hz (A), 2150
Hz (B) and 4300 Hz (C), respectively.
Harmonics in cosine phase.
Left panels: spatio-
temporal discharge pattern, displayed as a function of normalized time (horizontal axis) and
harmonic number (vertical axis). Right panels: Ravg (black) and MASD (red) derived from
the corresponding response patterns. Error bars correspond to ±+one standard deviation
obtained by bootstrap resampling of the stimulus trials (see Methods).
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4
Normalized Time (Number of Cycles)
Figure 2.5. Effect of stimulus level on the response pattern of one AN fiber (CF
Ravg
(black) and MASD
= ] 920
Hz).
(red) at 35 (top), 50 (middle) and 65 (bottom) dB SPL, respectively.
The threshold for a pure tone at CF was 25 dB SPL. Features as in Fig. 2.4.
80
478
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spatial derivative
3:
o
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1.5
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Normalized Time (Number of Cycles)
Figure 2.6. Solid lines show single-fiber models fit to profiles of Ravg (black) and MASD
(red) derived from the spatio-temporal response pattern of a AN fiber with CF of 2150 Hz
(same fiber as in Fig. 2.48).
Light shadings indicate the area between the top and bottom
envelopes of each fitted curve. Dark shadings correspond to two typical standard deviations
of the data points, derived from bootstrapping
(see Methods).
The ratio of these two
quantities is used as an estimate of the strength of the pitch cues provided by each pitch
representation.
81
--
Level re Threshold
20 - 39 dB
0- 20 dB
tfl. 20
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5
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Tuning Curve CF (kHz)
Figure 2.7. Relationships between estimates of CF derived from profiles of Ravg, MASD
and from pure-tone tuning curves. Filled circles show differences between estimates,
expressed as a percentage of the tuning-curve CF. Results are grouped by level relative to
each fiber's threshold for a pure tone at CF (columns). Crosses indicate the measurements
for which it was not possible to reliably estimate the CF from profiles of Ravg and MASD.
Solid lines indicate best linear fits(significant at the 0.01 level or better) across all levels.
Harmonics in cosine phase.
82
Level re Threshold
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4
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20 - 39 dB
dB
5
1
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':'
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•
1
Tuning
2
1
3
5
0
0.1
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"-r.
1
5
Curve CF (kHz)
Figure 2.8. Precision of CF estimation based on profiles of Ravg (black) and MASD
filled circles show
3
standard deviations of the MASD-
and Ravg-based
(red):
estimates of CF
(expressed as percentage of the tuning curve CF). Results are grouped by level relative to
each fiber's threshold at CF. Triangles indicate data points for which the SD was out of the
range defined by the vertical axes. Harmonics
in cosine phase.
83
level re Threshold
0-19
-
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200
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5
0
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1
3
5
,.
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100
0
0.1
:iJ.r
1
3
5
Tuning Curve CF (kHz)
Figure 2.9. Strength of rate-place and spatio-temporal cues to resolved harmonics.
circles show harmonic
and MASD
strengths computed
Filled
for best-fitting curves to profiles of Ravg (black)
(red). Triangles indicate data points for which the harmonic
strength was out of
the range defined by the vertical axes. Results are grouped by level relative to each fiber's
threshold for a pure tone at CF. Harmonics
in cosine phase.
84
level re Threshold
0-19
40+ dB
20 - 39 dB
dB
1
1
1
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0.5
CD
C
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en
-1
0.1
1
3
5
-1
0.1
1
3
5
-1
0.1
1
3
Tuning Curve CF (kHz)
Figure 2.10. Comparison
of the strengths of rate-based and spatio-temporal
Filled circles show normalized
9. Positive values mean
strength differences for the harmonic
greater strength for the spatia-temporal
representations.
strengths shown
in Fig
representation, and vice
versa. Results are grouped by level relative to each fiber's threshold for a pure tone at CF.
Red lines indicate best linear fits (significant at the 0.01 level or better) across all levels in
the 1-3 kHz CF-range.
Purple lines indicate best linear fits (significant at the 0.01 level or
better) across alllevels in the 3-5 kHz CF-range.
85
Harmonics
in cosine phase.
5
Cosine Phase
Schroeder Phase
Alternating Phase
560
0IL.
U::
u
630
720
-:- 3.5
CIl
.0
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840
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1008
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o
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340
1
2
3
4
0
2
0.5
Normalized
MASD
Normalized
R
1680
o
0.5
avg
4
3
Normalized Time (Number of Cycles)
Figure 2.11. Effect of phase relationship
patterns and on rate-place
among the harmonics
and spatio-temporal
representations.
for one fiber with CF at 2520 Hz for stimuli with harmonics
cosine) (B), and Schroeder
the respective
(C) phase.
The corresponding
maxima, are plotted in panels 0 and E.
86
on spatio-temporal
Response
response
patterns are shown
in cosine (A), alternating
Ravgs and MASDs,
(sine-
normalized
by
1
80
Q)
en
CO
or.
80
C)
c:
CI)
;: 40
CO
c: 40
;: 40
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fit •
40
20
60
en 100
co
or.
c. 80
Q)
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20
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40
0
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20 40
60 80 100 120
Cosi ne phase
20
40
60
80
Schroeder phase
120
CI)
en 100
co
or.
c. 80
100
en
co
or. 80
c.
60
Q)
c:
C)
0
80
120
Q)
60
60
Schroeder phase
120
c:
;:
co
c:
...
40
20
80
Cosine phase
-
-a 60
c.
c:
;;:
CO
CO 60
or.
C)
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en
rJ)
c. 60
80
Q)
Q)
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40
c:
;: 60
co
c: 40
20
<C
...
Q)
20 40
20
60 80 100 120
Schroeder
phase
20 40
60 80 100 120
Schroeder
phase
Figure 2.12. Effect of phase relationshipamong the harmonics on the strength of the spatiotemporal (A-C) and rate-place (D-F) cues to resolved harmonics.
comparisons of harmonic strengths for profiles of MASD
from spatio-temporal patterns of response to complex
(red) and Ravg (black) derived
tones with harmonics
alternating and Schroeder phase. CFs vary from I kHz to 3 kHz.
equality.
87
Filled circles show
in cosine,
Solid lines indicate
88
Chapter 3
Frequency Selectivity of Auditory-Nerve Fibers
Studied with Band-Reject Noise
89
3.1 INTRODUCTION
One of the most important functions of the cochlea is to mechanically separate sounds
into their frequency components. Models of auditory processing, used to predict a wide
variety of psychophysical phenomena, simulate the frequency analysis performed by the
cochlea by means of a bank of overlapping band-pass filters, referred to as "auditoryfilters".
The most popular technique in psychophysics for estimating the parameters of human
auditory filters is the "notched-noise method", introduced by Patterson (1976) and refined by
others (for example Patterson et al. 1982; Glasberg and Moore 1990; Rosen et al. 1998). The
notched-noise method consists of measuring the threshold level at which a band-reject noise
impedes the detection of a fixed-level pure-tone signal, as a function of the width of the
rejection band and its placement with respect to the tone. Since the ability of one sound (the
"masker") to interfere with the perception of another sound (the "signal") is commonly
known as "masking", such thresholds are commonly referred to as "masked thresholds".
Auditory filters are then estimated under the assumptions ("power spectrum model of
masking", Fletcher 1940) that (1) when detecting a signal in the presence of a masking
sound, the listener attends to the filter with the highest signal-to-noise power ratio at its
output, (2) detection thresholds correspond to a constant signal-to-masker power ratio at the
output of the filter, and (3) the filter is linear for the range of signal and masker levels used to
define it.
There is a general agreement about the fact that none of these assumptions is strictly true.
For example, psychophysical studies have shown that auditory filters are non-linear in that
their parameters depend on signal level (Moore & Glasberg 1987). Also, listeners can
combine information from more than one auditory filter in order to enhance signal detection
(Spiegel 1981; Buus et al. 1986).
These inconsistencies in the model do not mean that the basic concept of the auditory
filter is wrong.
On the contrary, this concept has proven to be very useful for studying a
wide variety of psychophysical
phenomena
[for example loudness (Moore et al. 1997),
"resolvability" of individual harmonics of complex tones (Bernstein 2006), certain aspects of
hearing impairment (Patterson et al. 1982)] and is a fundamental stage of models at the basis
90
of many useful applications (for example music or speech coding, noise reduction, hearing
aids).
Despite the importance of the notched-noise method, its physiological validity has never
been directly tested. In this study, we defined and measured masked thresholds of auditorynerve fibers in anesthetized cats using the same stimuli as in psychophysics, and compared
neural auditory filters derived by the notched-noise method to traditional measures of
frequency selectivity in auditory-nerve fibers such as pure-tone tuning curves.
A key question is whether the frequency selectivity observed in human psychophysics
experiments is primarily determined by peripheral tuning. We approached this issue by
developing a quantitative description of auditory filters parameters for individual fibers that
allowed us to constrain a neural population model.
The population model was used to
generate predictions of psychophysical notched-noise masked thresholds (largely unavailable
for the cat), which allowed us to test whether predicted psychophysical auditory filters match
the corresponding neural filters in the cat.
3.2 MATERIALS AND METHODS
Procedure
Methods for recording from auditory-nerve fibers in anesthetized cats were as described
by Kiang et al. (1965) and Cariani and Delgutte (1996a).
Healthy adult cats of either sex and free of middle-ear infection were used in these
experiments. Cats were anesthetized with Dial in urethane (75 mg/kg), with supplementary
doses given as needed to maintain an areflexic state. The posterior portion of the skull was
removed and the cerebellum retracted to expose the auditory nerve.
The ear canals were
transected for insertion of closed acoustic systems. The tympanic bullae and the middle-ear
cavities were opened to expose the round window. A silver electrode was positioned at the
round window to record the compound action potential (CAP) in response to click stimuli, in
order to assess the condition and stability of cochlear function. Throughout the experiment
the cat is given injections of dexamethasone (0.26 mg/kg) to prevent brain swelling, and
91
Ringer's solution (50 ml/day) to prevent dehydration. All animals remained anesthetized at
all times until death or euthanasia.
The cat was placed on a vibration-isolated table in an electrically-shielded,
temperature-
controlled, sound-proof chamber. Sound was delivered to the cat's ear through a closed
acoustic assembly driven by an electrodynamic
speaker (Realistic 40-1377).
The acoustic
system was calibrated to allow accurate control over the sound-pressure level at the tympanic
membrane.
Stimuli were generated by a 16-bit digital-to-analog
converter (Concurrent
DA04H) using sampling rates of 20 kHz, 50 kHz or 100 kHz, depending on CF. Stimuli
were digitally filtered to compensate for the transfer characteristics of the acoustic system.
Spikes were recorded with glass micropipettes filled with 2 M KCl. The electrode was
inserted into the auditory nerve and then mechanically advanced using a micropositioner
(Kopf 650). The electrode signal was bandpass filtered and fed to a custom spike detector.
The times of spike peaks were recorded with 1-ps resolution using custom circuits and saved
to disk for subsequent analysis.
A click stimulus at approximately 55 dB SPL was used to search for single units. Upon
contact with a fiber, a frequency tuning curve was measured by an automatic tracking
algorithm (Kiang et al. 1970) using 50-ms tone bursts, presented at a rate of 0l/s, and the
characteristic frequency (CF) was determined. The spontaneous firing rate of the fiber was
measured over an interval of 20s. The responses of the fiber to notched-noise stimuli were
then studied.
Stimuli
Stimuli were pure tones in Gaussian band-reject noise, with rejection bands ("notches")
placed both symmetrically and asymmetrically around the tone frequency.
The signals were
100-msec tone bursts at 0.5, 0.75, 1, 1.4, 2, 2.8, 4, 6, 8, 12, 16 kHz (an half-octave spacing),
whichever of these frequencies was closest to the CF of the fiber, estimated from the tuning
curve (although a one-octave spacing was used in some early experiments). The fixed signal
level (Rosen and Baker 1994) was chosen to be 10-15 dB above threshold at the tone
frequency.
symmetrically
The noises were band-reject
noises with the rejection band placed either
at the signal frequency, or centered 10% above or 10% below the signal
92
frequency.
These asymmetric notch placements are essential for deriving auditory filter
shapes without assuming that the filters are symmetric.
The width of the rejection band was
varied from 0% to 200% of the signal frequency, in steps of 20%. The upper passband of the
noise extended to 4 times the signal frequency.
One interval, containing both the tone and
the masker, alternated with one interval containing only the masker. The masker duration
was 124 ms, with cosine-square rise-fall time of 12 mns,and the signal occupied the central
100 ms of the masker (Figure 3.1A). The Gaussian random noises were synthesized by first
creating the desired spectrum in the frequency domain (real and imaginary part), and then
inverse Fourier transforming and windowing to the desired length.
ExperimentParadigms
For each notch condition, an automated tracking algorithm (PEST, Taylor and Creelman
1967) was employed to determine the threshold noise level which just masks the rate
response to the tone. Specifically, the number of spikes in response to signal in noise had to
exceed the spike count for the noise alone on 75% (threshold criterion) of stimulus
presentations for the signal to be detected.
At the end of the experiment, a cumulative
Gaussian distribution (error function) was fitted to the measured samples of the
"neurometric" function (percent correct as a function of masker level) by the maximumlikelihood method (Hall, 1968) to obtain a more reliable threshold estimate (Figure 3.2).
This "simultaneous masking" paradigm is comparable to those used in psychophysics.
One of the assumptions of the power spectrum model of masking is that masking occurs
only if the masker contributes significant energy at the output of the auditory filter tuned to
the signal frequency, thus "swamping" the response in the channel dedicated to the detection
of the signal. This type of masking is commonly referred to as "excitatory" masking.
However, there is evidence that a masker can "suppress" the response of an AN fiber to a
tone signal even if it does not, by itself, elicit any excitatory response in that fiber (Sachs &
Kiang 1968; Costalupes et al. 1984; Delgutte, 1990). Excitatory and suppressive masking are
not mutually-exclusive phenomena and can occur together (Costalupes et al. 1984; Delgutte,
1990). Excitatory masking dominates when the signal and the masker are close in frequency,
while suppressive masking is significant for masker frequencies far from the signal
frequency, particularly at high masker levels (Delgutte 1990a).
93
To estimate the contribution of suppressive masking, when possible thresholds were also
measured using a non-simultaneous, "pseudo-masking" paradigm, which differs from the
simultaneous masking paradigm described above in that the tone alone alternates with the
masker alone (Figure 3.1B). Masking per se is does not occur in this paradigm, because
signal and masker are sufficiently separated in time. However, non-simultaneous threshold
corresponds to the masker level that at which the response to the signal alone exceeds the
response to the masker by a just-detectable increment, effectively representing the threshold
that would be measured if masking were due exclusively to an excitatory mechanism.
Therefore,
the difference
between simultaneous
and non-simultaneous
threshold
is an
estimate of the contribution of suppression to simultaneous masking (Delgutte 1990a).
Single-fiberanalysis
Linear auditory-filter models were fit to single-fiber neural masked thresholds by
assuming that, for each fiber, threshold corresponds to a constant signal-to-masker power
ratio at the filter output for all notch widths (a "physiological adaptation" of the power
spectrum model of masking). A Rounded Exponential ("Ro-Exp") function (Rosen et al.
1998; Patterson et al.1982), was used to express the power spectrum of the auditory filters.
The general expression is the following:
(s -)
H(f) =(l-w)(l+p (
)e
2
(f - f.)
-f,.
~~~~(s
JH(f)2 =(1+ (ff))e
t(U, -sf)
+w(+t f))e
f
f.
forf<f
forf> fc
fc is the center frequency of the filter, while p and q independently control the low- and
high-frequency
slopes, respectively.
auditory filter is approximated
On the low frequency side of fo the "tail" of the
by a second, shallower rounded-exponential
term whose
weight and slope are controlled by w and t, respectively.
Fits provided by three different versions of this model (Figure 3.3) were compared: a
symmetric model without tail (w = 0; p = q), characterized by 2 free parameters (fc and p); an
asymmetric model without tail (w = 0; p # q), characterized by 3 free parameters (fc, p and q)
94
and a complete model (asymmetric with tail), characterized by 5 free parameters (f, p and q,
w and t). The best-fitting model was selected by minimizing the variance of the residuals,
which takes into account the number of degrees of freedom of each model.
"Bootstrap" resampling (Efron and Tibshirani 1993) was performed on the data recorded
from each fiber, in order to evaluate the statistical properties of the results and derive error
bars on the parameters of the fitted filters. In particular, for each notch width and noise level,
two-hundred resampled data sets were generated by drawing with replacement from the set
of original spike trains.
Auditory filters were fit independently to sets of thresholds
computed from each bootstrap data set.
Populationmodel
A neural population model was developed to allow a rigorous comparison between
simulated psychophysical and physiological auditory filters. Filter models for single fibers
cannot be compared directly with psychophysical auditory filters because of the lack of cat
behavioral masking data from experiments using notched noise. Predictions of behavioral
masked thresholds were generated, based on the assumption that behavioral threshold
corresponds to the best (highest) threshold in the entire simulated neural population
(Delgutte, 1990).
"Pseudo-psychophysical" auditory filters were predicted using the following procedure:
1. A neural filter population was simulated, consisting of 250 logarithmically equally
spaced center frequencies (CFs) over the range 500 Hz - 16 kHz.
The parameters of
the population filters at each CF were obtained by sampling uniformly within ±+one
standard deviation of the mean of the single-fiber model parameters for all fibers in an
octave-wide band including the CF.
2. Neural masked thresholds for a given tone signal were computed as a function of noise
condition for all the filters in the population.
Threshold was assumed to correspond to
a constant signal-to-noise ratio (SNR) at the output of each filter. For each filter, the
SNR at threshold
was set to the mean value for all fibers in the corresponding
frequency band.
3. For each noise condition, the predicted population masked threshold was defined as the
maximum masked threshold (meaning best performance) in the entire population.
95
While it is theoretically possible for an ideal processor to achieve better performance
than that of any single neuron (Green 1958, Siebert 1968), this simple rule has the
advantage of requiring no assumptions about correlation between the discharge patterns
of different neurons.
4. A pseudo-psychophysical
filter model was fit to the set of predicted population masked
thresholds using the same method as for the neural data.
The entire procedure was repeated 100 times to obtain measures of the variability of the
results.
3.3 RESULTS
Single-fiberresults
Our results are based on responses to tones in notched noise recorded from 47 auditorynerve fibers in 7 cats. Of these, 31 had high SR (> 18 spikes/s), 12 had medium SR (between
0.5 and 18 spikes/s) and 4 had low SR (< 0.5 spike/s). The CFs of the fibers ranged from 200
Hz to 23 kHz.
Figure 3.4 shows results for two auditory-nerve fibers with CFs of 1454 Hz (A) and 6775
Hz (B), respectively.
The frequency of the pure tone was 1000 Hz for the low-CF fiber and 8
kHz for the high-CF fiber. The tones were presented at 12 and 10 dB, respectively, above
threshold at the tone frequency. Masked thresholds (black) are plotted as a function of notch
width for three different placements of the notch center frequency with respect to the tone
frequency (see Methods).
For the low-CF fiber (A), a Ro-Exp model without tail gave
accurate predictions (orange) of masked thresholds. For the high-CF fiber (B), on the other
hand, a model without a low-frequency tail fit poorly (orange), while a complete model
including a low frequency tail (blue) was in this case necessary to capture the main trend in
the dependence of masked thresholds on notch configuration.
Panels C and D of Figure 3.4 show comparisons of the best-fitting neural auditory-filters
(orange and blue, as in A and B) to pure-tone tuning curves (black) measured from the same
fibers.
Note that these are not fits, but rather two independent measures of frequency
selectivity obtained with very different stimuli. The asymmetrical Ro-Exp filter without tail
96
(orange) crudely approximates the pure-tone tuning curve for the low-CF fiber (C), while
only a complete Ro-Exp model with tail (blue) can mimic the low-frequency portion of the
tuning curve of a high-CF fiber (D).
Figure 3.5A shows the overall errors (r.m.s., the square root of the mean square error) of
the best-fitting Ro-Exp models in predicting neural masked thresholds across all the fibers in
our sample.
R.m.s. errors were typically below 4 dB throughout the 0.2-23 kHz range of
CFs, while no obvious trend can be detected in the values of the errors as a function of fibers
CF. Figure 3.5B shows the fraction of the variance in the data that was accounted for by the
best-fitting Ro-Exp model for each fiber in our sample (R2 , or "R-square").
R-squares were
greater than 0.8 in 96% of the cases, indicating that linear filter models are applicable to
predict masked thresholds in notched noise of AN fibers in the cat.
Different versions of the Ro-Exp model gave the best fits to neural masked thresholds
depending on the CF range (Figure 3.6). For low-CF (< I kHz) fibers, as in the example of
Fig., 3.4 (A,C), asymmetrical models without a low frequency tail (orange) gave the best fits
in 12 of 15 cases. On the other hand, for high-CF (> 3 kHz) fibers, as in the example of Fig.
3.4 (B,D), a model with a low frequency tail (blue) was necessary to obtain the best fits in 13
of 15 cases. For CFs between 1 kHz and 3 kHz, any of the 3 models could give the best fit,
depending on the fiber.
Figure 3.7 shows that the center frequencies of the best-fitting model neural filters are in
very good agreement with the CFs estimated from the pure-tone tuning curves (correlation
coefficient = 0.994, p < 0.0001). This is a sanity check of the validity of our approach:
although auditory filters estimated with the notched-noise method are not constrained to have
the same center frequency as the CF of tuning curves obtained with pure tones, a large
discrepancy between the two would certainly raise some questions about the validity of the
method.
A commonly-used measure of sharpness of tuning for both the model filters and puretone tuning curves is the QERB. defined as the ratio of the center frequency (or CF) to the
equivalent rectangular bandwidth (ERB) (the ERB of a filter is defined as the bandwidth of
an ideal rectangular filter which passes the same amount of energy as the filter for white
noise inputs and has the same maximum gain). Figure 3.8 shows QERBS of model filters and
tuning curves (black diamonds for the tuning curves and blue for the Ro-Exp filters) as a
97
function of CF for single-fibers.
across 6 non-overlapping
QERBS
QERBS
octave-wide frequency bands centered at 0.5, 1, 2, 4, 8, and 16
Error bars span the interquartile ranges of the
kHz, respectively.
frequency bands.
Solid lines show the median values of the single-fiber
QERBS
across the same
of model filters (blue) and tuning curves (black) are very similar
throughout the entire frequency range and they increase systematically with CF, consitent
with previous results suggesting a progressive sharpening of the relative bandwidth of
cochlear filters with respect to their center frequency (Kiang et al. 1965; Shera et al. 2002).
Analysis of covariance was performed on the data of Fig. 3.8 to quantify whether there was a
significant dependence on CF, group (tuning curves or model filters), or a combination of
both. The result of this analysis showed that the
QERBS
increased significantly (p = 0) with
CF, and their values were fit by a straight line (red) on log-log coordinates whose slope (0.49
±+0.07)
and intercept were not significantly different for model filters than for tuning curves
(p= 0.63).
Effect of suppression
In order to quantify the contribution of suppression to masked thresholds and to evaluate
its effect on the parameters of auditory filters, we compared thresholds measured with the
simultaneous-masking and a non-simultaneous, "pseudo-masking" paradigm (see Methods).
Figure 3.9 shows results for a fiber with a CF of 1100 Hz. Thresholds measured with the
non-simultaneous paradigm (squares) are generally higher (meaning less masking) than those
obtained
with
the
simultaneous
paradigm
(diamonds).
Differences
between
non-
simultaneous and simultaneous thresholds are effectively estimates of the contribution of
suppression to simultaneous masking (Delgutte 990a). These differences correspond to the
additional amount of noise power needed to just mask the signal in the non-simultaneous
condition, designed to be equivalent to one in which masking were exclusively excitatory.
Non-simultaneous
thresholds also increase more rapidly with notch width than simultaneous
thresholds, consistent with observations that suppressive masking is increasingly significant
for masker frequencies farther from the signal frequency (Delgutte 1990a). Correspondingly,
the auditory filter estimated with the non-simultaneous pseudo-masking paradigm is sharper
than the one derived using the simultaneous masking paradigm.
98
This trend was observed for the majority of the fibers for which we were able to measure
masked thresholds in the two conditions at the same level (Figure 3.10). In 13 of 21 cases
(CFs ranging from 975 to 22834 Hz), the ERB of filters derived with the simultaneous
masking paradigm were significantly (p < 0.001) wider than those of non-simultaneous filters
(paired t-test on the sets of ERBs obtained over 100 bootstrap replications of the data for
each fiber).
Surprisingly,
in 4 cases (CFs of 340, 4950, 6775 and 22854 Hz; all high
spontaneous rate fibers), the opposite was true, i.e. the simultaneous filters was significantly
narrower than the corresponding non-simultaneous
ones. Finally, in 4 cases (CFs of 1450,
1520, 2052 and 4750 Hz, respectively; two high- and two medium- spontaneous rate fibers)
the ERBs obtained using the two paradigms were not significantly different.
Populationmodel and "pseudo-psychophysical"filters
Pseudo-psychophysical auditory filters were fit to sets of masked thresholds predicted
from a population of neural auditory filters (see Methods), for tone signals ranging from 750
Hz to 14 kHz. The shape of each of the population filters was selected to be consistent with
the trend observed for individual fibers (Figure 3.6): filters were asymmetrical without tail
below
kHz, asymmetrical with low-frequency tail above 5 kHz, while filters between 1 kHz
and 3 kHz were randomly selected among the three models.
In order to evaluate the
statistical properties of the results, one hundred populations of neural auditory filters were
generated (see Methods), and one example is shown in Figure 3.11. As for the neural data,
the same three versions of the Ro-Exp function were used to estimate the frequency response
of the pseudo-psychophysical auditory filters.
Figure 3.12 shows two examples of the results of this simulation, for tone frequencies of
750 Hz (A) and 8000 Hz (B), respectively.
In both cases the tone level was set at 20 dB SPL.
Predictions of behavioral masked thresholds (black) are plotted as a function of notch width
for three different placements of the notch center frequency with respect to the tone
frequency.
Purple solid lines show thresholds for the best fitting pseudo-psychophysical
auditory-filters. At the lower frequency (A), the best-fitting model was asymmetrical without
tail, while at the higher frequency (B), a model including a low frequency tail produced the
best fits. Panels C and D show the transfer function of the best-fitting filters.
99
In general, pseudo-psychophysical
filters produced excellent fits (1.3 - 4.8 dB r.m.s.
error, R-squares all greater than 0.98) to behavioral masked thresholds predicted from the
simulated neural populations throughout the 0.75 - 14 kHz frequency range. Similar to the
trend observed for neural filters (Fig. 3.6), asymmetrical filters without tail gave the best fits
for tone frequencies below 3 kHz, while filters with a low-frequency tail were necessary to fit
predicted behavioral masked threshold at frequencies above 3 kHz. The center frequencies
of the pseudo-psychophysical filters were very similar (median absolute deviation equal to
0.7%) to the frequencies of the corresponding tone stimuli.
Figure 3.13 shows how QERBS of pseudo-psychophysical auditory filters (purple)
compared to QERBS of neural auditory filters (blue). For the pseudo-psychophysical
filters,
error bars indicate the interquartile
neural
populations.
QERBS
ranges of the QERBS across 100 simulated
of the pseudo-psychophysical
filters are in line with those of neural
filters throughout the entire frequency range, the only small difference being at the lowest
CFs, where pseudo-psychophysical
corresponding neural filters.
filters appear to be slightly sharper (larger QERB) than the
These findings are similar to existing data comparing
behavioral and neural measures of frequency tuning using rippled and band-reject noise in
the guinea pig (Evans et al. 1991). We are unaware of the existence of behavioral estimates
of auditory filters using band-reject noise in the cat.
3.4 DISCUSSION
We measured masked thresholds of auditory-nerve (AN) fibers in anesthetized cats for
pure tones in band-reject noise at low stimulus levels, and compared neural auditory filters
derived by the notched-noise method to pure-tone tuning curves measured in the same fibers.
We found that linear Ro-Exp (rounded-exponential) filter models successfully predict the
dependence of masked thresholds on notch configuration for fibers of all CF (Fig. 3.4 and
3.5). Different versions of the Ro-Exp model gave the best fits to thresholds depending on
the CF: best-fitting filters were typically asymmetric for fibers with CFs below 1 kHz, while
only models including a low-frequency "tail" (mimicking the low-frequency portion of the
tuning curves) produced satisfactory fits for fibers with CFs above 3 kHz. For CFs between
100
I kHz and 3 kHz, different forms of the model could produce the best fit, depending on the
fiber.
The center frequencies of the model filters closely agreed with tuning-curve CFs (Fig.
3.7) despite the fact that the signal frequency did not exactly match the CF and could be as
far as half an octave away from the CF). This result supports the robustness of the notched
noise method, since signals away from the CF could generate suppressive effects on masker
components at frequencies close to the CF (particularly for narrow notch widths) (Delgutte
1990a) that are not consistent with the assumptions of the power spectrum model of masking.
Measures of sharpness of tuning (QERBS, the ratio of CF to Equivalent Rectangular
Bandwidth) of tuning curves and neural filters were also in good agreement (Fig. 3.8). QERBS
of both neural filters and tuning curves increased systematically with CF, consistent with the
progressive sharpening of peripheral tuning with increasing CF, and this increase was well fit
by a power function of CF with an exponent of 0.49 ±+0.07. The 95% confidence interval for
this parameter overlaps with both that for the 0.37 ( 0.10) exponent found by Shera et al.
(2002) (for the CF dependence of Ql0 in pure-tone tuning curves from AN fibers in the cat)
and that for the 0.37 (
0.09) exponent found by Cedolin and Delgutte (2005) (for the
increase in resolvability of harmonics of complex tones with CF in the cat AN - see Chapter
1).
It is interesting to notice that, despite the highly nonlinear nature of the cochlea, there is
broad agreement among values and trends observed in estimates of frequency selectivity (at
least at low sound levels) of cat AN fibers obtained using very different stimuli and methods:
pure-tone tuning curves (Kiang et al. 1965; Shera et al. 2002), broadband-noise reversecorrelation techniques (de Boer and de Jongh 1978; Carney and Yin 1988), synchrony to
pairs of tones (Greenberg et al. 1986), rippled noise (Evans & Wilson 1973), complex tones
(Cedolin and Delgutte 2005) and masking by notched-noise (present study).
One important aspect of peripheral frequency selectivity that was not addressed in our
study is its level dependence.
We typically were not able to "hold" fibers long enough to
obtain data at different levels. However, even if holding time were not an issue, measuring
thresholds at high levels (which would lead to an inaccurate assessment of cochlear
frequency selectivity) would be hard because of rate saturation, especially for fibers with
high-SR, due to their limited dynamic ranges. An alternative strategy to compare frequency
101
tuning at different levels would be looking at data obtained from low-threshold, high-SR
fibers at low levels and from high-threshold, low-SR fibers at high levels (ideally in the same
animal). The very small number of fibers with high threshold and low spontaneous rate in
our sample prevented us from performing this comparison.
To quantify the contribution of suppression to masked thresholds and to evaluate its
effect on the model filters parameters,
we compared
thresholds
measured
with the
simultaneous-masking and a non-simultaneous, "pseudo-masking" paradigm, designed to be
equivalent to one in which masking is exclusively excitatory.
For a majority of fibers (Fig.
paradigm were better than those
3.9), thresholds measured with the non-simultaneous
obtained with the simultaneous paradigm, suggesting a contribution of suppression to
simultaneous masking (Delgutte 1990a). Differences in threshold, and therefore the relative
amount of suppressive masking, increased with notch width, consistent with previous results
with pure tones (Delgutte 1990a) showing large suppression for masker frequencies farther
from the signal frequency.
filter estimated with the non-
In these cases, auditory
simultaneous pseudo-masking paradigm were somewhat sharper than those derived using the
simultaneous masking paradigm (Fig. 3.10). This result suggests that frequency selectivity
estimates derived in psychophysics using forward masking paradigms (e.g. Glattke 1967;
Houtgast 1972; Moore 1978), designed to minimize the effect of suppression, may be more
accurate than those derived using simultaneous
masking paradigms.
For some fibers,
auditory filters bandwidths estimates with the two paradigms were not significantly different,
indicating weak or no suppression, and in 4 of 21 cases, simultaneous filters were sharper
than non-simultaneous filters. Interestingly, the only CF-range consistently showing an effect
of suppression was the one between 600 Hz and 2 kHz (Fig. 3.10), in contrast with the
findings of previous studies showing weak suppression for low-CF fibers compared to highCF fibers in cat (Delgutte 1990b), chinchilla (Harris 1979) and gerbil (Schmiedt 1982).
An
important
issue
is whether
the
frequency
selectivity
observed
in human
psychophysical experiments is primarily determined by peripheral tuning or whether it is
sharpened by some more central neural mechanism.
We approached this issue by using
typical auditory filters parameters for individual fibers to simulate the performance of a
neural population in the detection of pure tones in notched noise. Specifically, the population
model was used to generate predictions of psychophysical
102
masked thresholds in notched
noise based on the simplifying assumption that, for each noise condition, the predicted
population masked threshold corresponds to the maximum masked threshold (meaning best
performance) in the entire population. By fitting "pseudo-psychophysical" auditory filters to
masked thresholds based on the population model, we were able to compare the predicted
psychophysical tuning sharpness to that of the underlying neural filters in each frequency
region. The results of this analysis suggest that the two are indeed similar at low levels, thus
supporting the use of the notched noise method to measure tuning sharpness from
psychophysical masking data.
3.5 CONCLUSIONS
We tested the physiological basis of the notched-noise method, widely used in
psychophysics to estimate peripheral frequency selectivity. Neural auditory filters were fit to
masked thresholds for pure tones in band-reject noise.
Estimates of neural frequency
selectivity derived using the notched-noise method show an increase in tuning sharpness with
CF that is similar tlothat obtained in previous studies using a variety of different stimuli and
techniques and is also in broad agreement with that inferred from results related to harmonic
resolvability in Chapter 1. A neural population model was used to test the extent to which
auditory filters estimated psychophysically match the underlying neural filters.
The
similarity observed between the frequency selectivity measured physiologically and the
predicted psychophysical tuning sharpness supports the use of the notched-noise method in
human psychophysics.
103
A
U)
::s
::s
E
..
en
RESPONSE~
INTERVALS
S+M
100
0
~
M
200
300
~
400
Time (ms)
B
-
en
RESPONSEINTERVALS
~
S
o
~
100
200
M
300
400
Time (ms)
Figure
3.1:
simultaneous
Gaussian
Simultaneous
masking
(A) and Non-Simultaneous
paradigm,
and interval
noise masker (red) alternated
non-simultaneous
masking
the first interval contained
with cosine-square
rise-fall
paradigm,
(B) masking
containing
both a pure tone (black)
with one interval containing
designed
only the pure tone.
to quantify
The duration
104
only the masker.
the contribution
In the
and a
In the
of suppression,
of each noise burst was 124 ms,
time of 12 ms, and the signal occupied
masker.
paradigms.
the central
100 ms of the
•
•
fn
CI)
E
::E
"Z
'01\
I: ::E
.2 c1;
0.75
0.5
•
•
Z
'U
ca
~
u.
o
-2
o
2
4
6
8
10
12
14
16
18
Masker level (dB)
Figure 3.2. Threshold determination.
During the experiment, a PEST tracking algorithm
(Taylor and Creelman 1967) was employed to determine the threshold (red cross) for each
notch condition. The number of spikes in response to signal in noise had to exceed the spike
count for the noise alone on 75% (green) of stimulus presentations.
At the end of the
experiment, a cumulative Gaussian distribution (error function, black solid line) was fitted to
the measured samples of the neurometric function by the maximum-likelihood
method (Hall,
1968). An estimate of the threshold error was determined by calculating the mean error ME
between the cumulative Gaussian and the measured samples of the neurometric function, and
determining the two masker levels (red circles) at which the cumulative Gaussian assumed
values equal to 0.75
:t
ME.
105
A
o
Symmetrical Model
Asymmetrical Model
B
m-20
--"C
_-40
~-60
\
-80
100
1000
10000
....
Asymmetrical
c ovarYing
tlp2tall ratio
m-2O
Model WITH TAIL ...
varYing tall bandwidth
0
---
-20
"C
_-40
-40
:J: -60
-60
-80
-80
-
100
10000
1000
100
Frequency
Figure 3.3. A: symmetric
model
Ro-Exp
asymmetric
Ro-Exp
without
(asymmetric
with low-frequency
model without
tail (w
=
tail).
106
10000
1000
(Hz)
low-frequency
0; p
i- q).
tail (w
C: complete
=
0; p
=
Ro-Exp
q).
B:
model
Figure 3.4 (next page):
Masked thresholds (black circles) and the fits provided by the
asymmetrical Ro-Exp models with (blue) and without (orange) low-frequency tail. A, C:
fiber CF = 1454 Hz; tone frequency = 1 kHz; tone level = 12 dB above threshold at 1 kHz.
B, D: fiber CF = 6775 Hz; tone frequency = 8 kHz; tone level = 10 dB above threshold at 8
kHz.
A, B: thresholds and fits are plotted as a function of notch width (expressed as a
percentage of the tone frequency) in three separate panels for notches placed symmetrically
and asymmetrically (10% above and 10% below) with respect to the tone frequency. C, D:
comparisons between the pure-tone tuning curves (black) and the best-fitting Ro-Exp
models with (blue) and without (orange) low-frequency tail.
107
A
Noise Fe
= 1000Hz
Noise Fe
= 1100Hz
Noise Fe
50
50
50
40
40
40
"0
~
30
30
30
...
20
20
20
10
10
10
m
= 900Hz
~
"0
C/)
G)
~
....
0
C
100
200
0
50
100
150
Notch Width %
0
50
100
150
0
m
-20
~ -40
:::=::
~
-60
-80
100
1000
Noise Fe
B
= 8000Hz
30
m
20
"0
"0
~
10
...
0
~
....
= 7200Hz
Noise Fe
20
20
10
0
0
-10
-10
-20
-20
D
0
20
40
60
80
0
50
Notch Width %
100
0
50
0
m
-20
~ -40
:::=::
~
= 8800Hz
30
40
~
C/)
G)
10000
Noise Fe
-60
-80
100
1000
Frequency
108
10000
(Hz)
100
10
. '''
' ''
)
9
8
m
7
6
0
6
E
03
1-
~~
~
~
~
o
2
0
lb
n00
10
· -~~~~~~~
0
·
so*
Oem~~~~
0
:.
.
·
_
.
10000
1000
100
I~~~~~~~~0I
0.9
00
0
0.8-
''
0
0
0.7
0.6
R2
0.5
0.4
0.3
0
0.2
0.1
nv
.
.I
1oo
,
.
10000
1000
CF (Hz)
Figure 3.5. A: R.m.s. error (dB) between measured masked thresholds and Ro-Exp model
predictions, plotted as a function of fiber's CF. B: R2 of the best fitting Ro-Exp models,
plotted as a function of fiber's CF.
109
14
I..w
..
..
12
symmetrical model
asymmetrical model
asymmetrical model with tail
10
8
6
4
2
0
CF < 1kHz
1 kHz < CF < 3 kHz
Figure 3.6. Best-fitting Ro-Exp model as a function of fiber's CF.
110
CF > 3 kHz
S
0
0
10000
0
I
N
1V
'10
0- r
a go
-0
0 U.
OU.
1000
0 I0
C
c)
0.
.0
0
inn
.. B
II .l
UV
_
100
1000
10000
Tunina Curve CF (Hz)
Figure 3.7. Center frequency of the best-fitting Ro-Exp model as a function of tuning curve
CF.
111
40
30
20
10
• •
m
a:
•
w
0
4
3
2
300
500
1000
2000
3000
Characteristic
Figure 3.8.
as a function
QERBS
QERBS
of tuning curves (black diamonds)
of fibers tuning-curve
across 6 non-overlapping
16 kHz, respectively.
5000
CF.
Medians
octave-wide
10000
20000 30000
Frequency (Hz)
and model neural filters (blue squares)
and interquartile
frequency
ranges (errorbars)
bands centered
of the
at 0.5, I, 2, 4, 8, and
Red line indicates best linear fit across both groups based on analysis
of covariance.
112
Noise Fe
_
= 1100Hz
50
50
40
40
'tJ
40
'0
s::.
30
II)
...
s::.
Noise Fe = 900Hz
60
m
~
Noise Fe = 1000Hz
30
II)
t-
20
20
20
10
10
0
50
o
100
100
200
Notch Width %
o
50
100
150
0
m
~
-40
~
-60
-
-
-20
Simultaneous
Masking
Non-simultaneous
Masking
Tuning Curve
-80
100
Figure 3.9.
Top:
(squares)
thresholds
threshold
at CF).
fitting
Ro-Exp
paradigm
10000
1000
Frequency
companson
measured
Bottom:
between
models derived
simultaneous
for the same fiber
comparison
(Hz)
between
(CF
113
=
and
non-simultaneous
1100 Hz, tone level
the pure-tone
with the simultaneous
(see Methods).
(diamonds)
tuning
curve
= 20
dB above
and the best-
(blue) and the non-simultaneous
(cyan)
•
'-
co
~
1000
••
0)C N
-
•
•
co~
u.J:
Q)a:-c
I
- '3:
C-c
~ C
CO CO
••
.~
.
• ••• •
•
•
,~ CO
~
C"
W
•
•
•
•• •
I
••
•
•
•
•
•
100
100
1000
10000
Characteristic
Figure 3.10. Comparison
of the equivalent
models derived with the simultaneous
Frequency (Hz)
rectangular
bandwidths
(blue) and non-simultaneous
114
of best-fitting
(red) paradigm.
Ro-Exp
o
-20
m
'C
-z
:::: -40
-60
1000
100
3000
10000
Center Frequency (Hz)
Figure 3.11.
Example of a population of neural auditory filters, with center frequencies
ranging from 500 Hz to 16 kHz. The shape of each of the population filters was selected to
be consistent
with the trend observed
for individual
fibers (Fig. 3.6): filters were
asymmetrical without tail (orange) below 1 kHz and asymmetrical with low-frequency tail
(blue) above 5 kHz, while between 1 kHz and 3 kHz filters were randomly selected among
the three models.
115
Figure 3.12 (next page):
Masked thresholds based on a neural population (black circles)
and the fits of the pseudo-psychophysical
auditory filters (purple). A, C: tone frequency =
750 Hz; tone level = 20 dB SPL. B, D: tone frequency = 8 kHz; tone level = 20 dB SPL.
A, B: thresholds and fits are plotted as a function of notch width (expressed as a percentage
of the tone frequency)
in three separate panels for notches placed symmetrically
and
asymmetrically (10% above and 10% below) with respect to the tone frequency. C, D: best-
fitting pseudo-psychophysical filters (asymmetrical without (C) and with (D) low-frequency
tail).
116
A
Notch Fc = 750
50
m
Notch Fc = 675
50
30
30
"0
s;
G1 20
~
40
20
30
....
10
"0
II)
20
s;
10
0
C
= 825
60
40
40
~
Notch Fc
50
100 150
10
0
50 100 150
Notch Width %
0
50
100
150
0
-20
m
~ -40
::=::
-
~ -60
-80
Notch Fc = 8000
B
104
103
Frequency (Hz)
2
10
Notch Fc = 8800
Notch Fc = 7200
40
40
40
20
"0
s;
20
20
0
0
0
-20
0
-20
0
-20
0
m
~
"
II)
G1
~
s;
....
0
m
50
100 150
50 100 150
Notch Width %
50
100
0
-20
~ -40
::=::
-
~ -60
-80
102
103
Frequency (Hz)
117
4
10
150
40
30
20
10
lD
a:
w
o
4
3
2
300
1000
500
2000
3000
Characteristic
Figure 3.13.
interquartile
bands
Blue:
interquartile
of model
ranges (error bars) of the
centered
psychophysical
QERBS
at 0.5,
model
], 2, 4, 8, and
filters
ranges of the
QERBS
across]
filters
as a function
across 6 non-overlapping
] 6 kHz,
as a function
20000 30000
10000
Frequency (Hz)
neural
QERBS
5000
respectively.
of their center
00 simulated
] ]8
of CF.
octave-wide
Purple:
frequency.
populations.
Medians
and
frequency
QERBS
of pseudo-
Error
bars show
Chapter 4. Summary and Conclusions
The main focus of this thesis was to investigate possible neural representations of the
pitch of harmonic complex tones, with particular attention devoted to discussing how well
pitch cues provided by each representation can account for major trends in human
psychophysical data on pitch perception.
In psychophysics, these trends are often
interpreted in terms of the extent to which harmonics are "resolved" (identified and/or
processed individually) thanks to the decomposition of incoming sounds into their
frequency components performed by the auditory periphery.
Given the possible
differences between the peripheral tuning sharpness in humans and in experimental
animals, we focused on estimating peripheral frequency selectivity and defining harmonic
resolvability in the species studied (in our case, the cat), thereby providing quantitative
grounds for discussing the results in terms of their implications for human psychophysics.
Here, we provide a summary of our main findings.
In Chapter 1, we studied harmonic resolvability based on average-rate responses of AN
fibers
and
investigated
two
traditional
representations
of pitch:
(1) a rate-place
representation of pitch, based on matching "harmonic templates" to the pattern of
excitation of an array of tonotopically-organized neurons, and (2) a temporal representation
of pitch, based on matching "periodic templates" to the most common interspike intervals
of a population of AN fibers.
We found evidence for an increase with CF of harmonic resolving "power" of single
AN fibers (expressed as
Nmax,
the maximum resolved harmonic number CF/F0), which
closely mirrored the increase with CF of tuning sharpness (expressed as a Q-factor, the
ratio CF/Bandwidth) for pure-tone tuning curves (Kiang et al. 1965; Liberman 1978; Shera
et al. 2002). An increase of Nmaxwith CF is also consistent with previous results obtained
in the AN with rippled noise (Wilson and Evans 1971) and in the AVCN with complex
tones (Smoorenburg and Linschoten 1977).
Also consistent with existing AVCN data
(Smoorenburg and Linschoten 1977), we found that Nmax decreases rapidly with small (1020 dB) increases in stimulus level, a phenomenon
that we attributed mainly to rate
saturation rather than to broadening of cochlear tuning with level.
119
We were consistently able to estimate pitch by matching harmonic templates to profiles
of average discharge rate against CF for F0s above 400-500 Hz. Below this lower limit,
larger errors became increasingly common, although some reliable estimates were obtained
for F0s as low as 250 Hz. Both the precision and the predicted strength of rate-based pitch
estimates increased with F,
consistent with an increase in the degree of harmonic
resolvability due to the progressive sharpening of cochlear tuning with CF. However, since
the predicted strength of rate-based pitch estimates continued increasing beyond 1400 Hz,
we concluded that a purely rate-place representation cannot account for the existence of an
upper limit to the perception of the pitch of missing-F0 harmonic complex tones in humans.
Rate-based pitch estimates were largely independent of the relative phases among the
harmonics, in agreement with results of human psychophysics experiment for complex
tones containing resolved harmonics (Houtsma and Smurzynski 1990; Shackleton and
Carlyon 1994). Given the rapid saturation of average-rate responses of the majority of the
AN fibers with stimulus
level, we deemed
it unlikely that a strictly
rate-based
representation can account for the (relatively) stable human performance in detecting small
F0-changes at high levels in the presence of resolved harmonics (Bernstein and Oxenham,
2006)
Pitch estimates derived by matching periodic templates to pooled all-order interspikeinterval distributions were highly accurate and precise for F0s up to 1300 Hz. Above this
upper limit, the purely temporal representation
broke down, arguably
due to the
degradation of phase-locking to the frequencies of individual harmonics. Thus, a pitch
representation based on interspike-interval distributions roughly predicts the upper limit for
the pitch of the missing fundamental in humans. However, the predicted strength of
interval-based pitch estimates was highest for F0s below 400-500 Hz (where harmonics are
poorly resolved, based on rate-results), and decreased monotonically with Fs (as the
degree of harmonic resolvability increases).
This observation led us to conclude that a
temporal representation cannot account for the greater salience of the pitch based upon
resolved harmonics compared to that based on entirely unresolved harmonics (Bernstein
and
Oxenham
2003b;
Houtsma
and
Smurzynski
1990).
The interspike-interval
representation was consistent with phase invariance for the pitch of stimuli containing
resolved harmonics, but could not completely_account for the perceptual dependence of
120
pitch on the relative phase among the harmonics when these are all unresolved (Lundeen
and Small 1984; Shackleton and Carlyon 1994).
Given the limitations of both a strictly place and a strictly temporal representations, in
Chapter 2 we tested an alternative spatio-temporal representation of the pitch of harmonic
complex tones, where cues to the frequencies of resolved harmonics arise from the spatial
pattern in the phase of the phase-locked response of AN fibers (Shamma 1985).
This
spatio-temporal representation was investigated in single AN fibers and the results were
first analyzed as a function of CF, and then re-interpreted as a function of FO by applying
the scaling invariance principle of cochlear mechanics (Zweig 1976). We found that the
spatio-temporal representation is viable in the cat for FOs between 300 Hz and I kHz.
We attributed the failure of the spatio-temporal representation below 300 Hz to poor
harmonic resolvability, and this lower limit was in broad agreement with that found for the
rate-place representation in Chapter 1. Although harmonic resolvability, as measured using
a strictly "spectral" definition like the one we adopted in Chapter 1, increases with FO, the
effectiveness of the spatio-temporal representation decreased very rapidly with FO above 1
kHz, a finding which we attributed to the rapid degradation of phase-locking
harmonic frequencies.
to the
The existence of an upper limit for the spatio-temporal
representation of pitch is thus in broad agreement with the upper limit for the pitch of
missing-FO complex tones in humans.
For FOs between 300 Hz and 1 kHz, a range in
which the spatio-temporal and the rate-place representation are both effective, we observed
a degradation of both representations with stimulus level. However, spatio-temporal pitch
cues remained up to 2-3 times stronger than rate-place cues at very high levels, suggesting
that the spatio-temporal representation may be more consistent with psychophysical data
than the rate-place representation. Finally, the spatio-temporal representation was also
consistent with human psychophysical data on phase-independence for pitch based on
resolved harmonics.
In Table 2, we present a summary of the different ranges of viability for the three pitch
representations examined in this thesis, and of their predicted adequacy in accounting for
psychophysical
data. Also presented in Table 2 is a speculation on where the FO-limits
observed for each representation in the cat would translate to in humans, based on the
assumptions that 1) cochlear filters are three times sharper in humans than in cats (Shera et
121
al. 2002) and that 2) the degree of neural phase-locking
and its upper limit are not
significantly different in the two species.
In Chapter 3, we studied peripheral frequency selectivity by applying the notched-noise
method, widely used in psychophysics, to single AN fibers. We found that, at low stimulus
levels, this method yields accurate predictions of single AN fibers' "masked" thresholds for
pure tones in band-reject ("notched") noises, across the entire range of CFs. We observed a
similar increase in tuning sharpness (expressed as a Q-factor, the ratio CF/Bandwidth) with
CF for both pure-tone tuning curves and neural auditory filters estimated with the notchednoise method. This increase was well-fit by a power function of CF with exponent equal to
0.49 ±+0.07, a value slightly higher, but not significantly different from those derived in
other studies using different experiment paradigms [0.37 ±+0.10, observed by Shera et al.
(2002) for the CF dependence of Ql0 in pure-tone tuning curves from AN fibers in cats;
0.37 + 0.09, found in Chapter 1 of the present study for the increase in Nmax with CF]. To
quantify the effect of suppression on estimated tuning sharpness, we compared the
equivalent rectangular bandwidths (ERBs) of neural auditory filters derived using a
simultaneous masking and a non-simultaneous masking paradigm. Although in the majority
of the cases suppression contributes to an overestimation of the ERBs, in a few cases this
trend is reversed, a finding for which we have no plausible explanation.
pseudo-psychophysical
auditory filters to masked thresholds predicted
population of neural auditory filters.
We also fit
from a model
The results showed a close agreement between the
predicted sharpness of psychophysical tuning and the frequency selectivity measured
physiologically,
thus supporting
the use of the notched-noise
psychophysics.
122
method
in human
Phase-
Predicts the
Predicted
FO
FO range
range
(Hz)
(Hz)
for cat
for humans
Depends on
upper limit
Robust
invariant
resolved
for pitch of
at high
for
harmonics
the missing-
levels
resolved
harmonics
FO
Rate-Place
InterspikeInterval
SpatioTemporal
-> 450
-<1300
300-
k
> 150
yes
no
no
yes
< 1300
no
yes
yes
yes
100- lk
yes
yes
partly
yes
Table 2: F0-ranges of viability for the rate-place, the interspike-interval and the spatiotemporal representations, and their predicted adequacy in accounting for psychophysical data.
123
Chapter 5. Implications and future directions
The results of our experiments indicate that three types of cues to the pitch of harmonic
complex tones are available in the auditory nerve: "rate-place" cues, reflecting harmonicity
of the frequency spectrum, "temporal" cues, reflecting periodicities in the time waveform
and "spatio-temporal" cues, reflecting modulations in temporal information with cochlear
place.
A fundamental question is whether the formation of a pitch percept is based upon
the exclusive use of any one of these three possible cues, or whether different mechanisms
are employed to different degrees.
Resolved
harmonics
are almost always available
to normal hearing listeners in
everyday acoustic environments, and pitch based on resolved harmonics is far more salient
than the pitch based on exclusively unresolved harmonics (Houtsma and Smurzynski 1990;
Bernstein and Oxenham 2003b). Recent psychophysical data (Bernstein and Oxenham
2006) shows a correlation between the degraded cochlear frequency selectivity of hearing
impaired listeners with sensorineural hearing loss and their deficit in F-discrimination
performance. Abnormally high F0 difference limens are also exhibited by cochlear implant
users, and their performance becomes very poor above 300 Hz (Zeng 2002; Shannon
1983), presumably due to the insufficient "place" information available from electrical
stimulation.
These observations
strongly suggest that a pitch code which privileges
resolved harmonics over unresolved harmonics (i.e. rate-place or spatio-temporal) is most
likely to be used in normal listening conditions. While a strictly rate-place representation
degrades rapidly with level and cannot account for the existence of an upper limit to the
human
perception
of the pitch
of the missing-F0
(Chapter
1), a spatio-temporal
representation may overcome this limitation (Chapter 2), thereby becoming the strongest
candidate to explain several major human psychophysical phenomena.
On the other hand, a weaker pitch can be heard by hearing-impaired
and normal-
hearing listeners also in the presence of only unresolved harmonics, a finding that cannot
be accounted for by any pitch representation dependent upon harmonic resolvability and
that therefore is in favor of the hypothesis that a purely temporal pitch representation must
indeed be used when place or (spatio-temporal) cues are unavailable.
A purely temporal
representation is also thought to be the most likely explanation for the weak pitch percepts
124
of cochlear implant users, because of the poor spatial resolution of the electrical
stimulation. In contrast, neural responses to electric stimulation with sinusoids and pulse
trains show a high degree of phase-locking, compared to that observed in response to
acoustic stimuli (ynes
and Delgutte 1992; Javel and Shepherd 2000). However, despite
the large amount of available temporal information, pitch is highly ambiguous and is heard
only at very low FOs, consistent with only a small contribution of a temporal mechanism to
pitch perception in normal conditions.
Another important issue is the fate undergone at more central stages of the auditory
pathway by the cues provided at the level of the AN by the different pitch representations
examined.
Spatio-temporal
pitch cues may be converted into place cues at some more
central stage along the auditory pathway.
The conversion of spatio-temporal cues into
place cues requires a neural mechanism 1) receiving inputs originating from neighboring
cochlear locations and 2) sensitive to differences in the relative timings of its inputs.
Although there is some evidence in the AVCN for neurons that may satisfy these
requirements
(Carney 1990), more data are needed to shed light on the exact neural
mechanisms (coincidence detection, lateral inhibition or a combination of the two are
proposed explanations) performing this operation.
Given the improvement in neural
synchronization exhibited by some AVCN cells (Joris et al. 1994), which might actually
enhance spatio-temrnporalcues to pitch, another possibility is that the extraction of these
cues takes place at a more central auditory center.
The generation of a pitch percept from a spatial pattern of activation, whether originally
generated by a purely rate-place code or produced by a transformation of spatio-temporal
cues into place cues, is generally assumed to be based on a harmonic template matching
mechanism. While there is strong evidence for a robust tonotopic mapping of the frequency
spectrum of sounds at virtually all levels of the auditory pathway (including the auditory
cortex) (Popper and Fay 1992), the process underlying the formation of preferential links
among harmonically-related channels is a matter of ongoing debate. A classical hypothesis
is that harmonic templates arise as a consequence of the frequent exposure to speech and
natural harmonic sounds in early development (Terhardt 1974). However, young infants
(7-8 months old) not only can perceive the pitch of the missing-F0, but also perform better
in the presence of low-order, presumably resolved harmonics (Clarkson and Rogers 1995),
125
suggesting that the formation of the harmonic templates may occur before a sufficient
exposure to sounds characterized by a rich harmonic structure.
An appealing alternative
theory has been proposed (Shamma and Klein 2000) according to which harmonic
templates arise from the exposure to any kind of broadband stimulation and not necessarily
to harmonic sounds. According to this model, the half-wave rectification performed by the
hair cells results in a high likelihood for highly-correlated
spike trains at harmonically-
related CFs, which are in turn reflected in the accumulation of coincidences over time in
these pairs of channels. The accumulation of coincidences between the firings in cochlear
channels that are harmonically-related
can conceivably explain the emergence, over time,
of harmonic templates for all FOs whose resolved harmonics are in the frequency range of
phase-locking.
The nature of this hypothetical mechanism underlying the formation of harmonic
templates is related, but not identical to the cues to resolved harmonics generated by the
variations in phase of the basilar membrane motion with cochlear place discussed in
Chapter
2.
Both mechanisms
are "spatio-temporal"
in that they depend
upon a
combination of place information and neural phase-locking. Both are also based on across-
CF interactions, but while the harmonic template generation process relies on correlations
between channels with wide separations, extraction of spatio-temporal cues to resolved
harmonics only requires comparisons between spike trains in neighboring channels. Crossfrequency coincidence detection is the most likely operation at the origin of the emergence
of the harmonic templates, while on the other hand both coincidence detection and lateral
inhibition (the latter effectively implemented in Chapter 2) could conceivably underlie the
extraction of spatio-temporal cues to resolved harmonics.
Although evidence for precise timing information (of the order of microseconds) like
the one needed for the preservation
of an interspike-interval
or a spatio-temporal
representation is weaker and weaker as one ascends the auditory pathway, phase-locking up
to 1 kHz has occasionally been observed in the medial geniculate body of the thalamus
(Rouiller et al. 1979) and in the cortex (de Ribaupierre et al. 1972). The existence of a
central pitch code entirely or partially based on temporal information cannot therefore be
completely ruled out. It has also been hypothesized that an interval code in the AN and CN
could be converted into some other types of temporal code, possibly on different time
126
scales or even asynchronous (Cariani and Delgutte 1996b) or distributing the large amount
of temporal information progressively more sparsely over a greater number of neurons
(Cariani 1999). A more realistic hypothesis is that even a purely temporal peripheral code
based on interspike intervals may somewhere be converted into a place code (Licklider
1951).
A physiologically-realistic model has recently been proposed (Wiegrebe and
Mecldis 2004), according to which a temporal representation of pitch in the AN may be
converted into a place representation by a population of units with sustained-chopper
responses in the VCN, albeit only for low FOs (below 500 Hz).
Given the important role played by pitch in source segregation in speech and auditory
scene analysis, the extent to which the proposed pitch mechanisms can represent more than
one complex tone is of great interest.
If based on a place code, the identification of two
simultaneous pitches would require an additional amount of frequency resolution, while if
based on a temporal code, segregation of two pitches would rely on simultaneous cues to
different periodicities.
Recent data from cat AN (Larsen et al. 2005) indicate that the rate-
place and the interval representation may be complementary
in their ability to represent
pairs of complex tones with a difference in F0 as small as 7%: the interval representation is
more effective at lower FOs and the rate-place
representation
at higher FOs.
This
performance is more than sufficient to account for the lower psychophysical limit for
simultaneous vowels of a 15-25% F-separation
which
human listeners
cannot
correctly
(Assmann and Paschall 1998), below
identify both pitches
simultaneously.
Unfortunately, data are not yet available for the spatio-temporal representation, where this
task could be problematic given the additional constraints imposed by the presence of two
complex tones: frequency resolution of two sets of harmonics and competition between
phase-locking to different, non-harmonically related frequencies. A systematic study of the
human perception of simultaneous complex tones would also be of high significance, since
sounds present in natural environments are very rarely encountered in total isolation.
Finally, a spatio-temporal
which we propose to be the basis for the
mechanism,
perception of pitch, could also in principle underlie the detection of pure tones (in the
frequency range of phase-locking) in the presence of masking sounds, and has also been
proposed as the basis for level discrimination
(Heinz et al. 2001). This idea might have
important implications for current methods used in psychophysics to estimate peripheral
127
frequency selectivity, which are all based on the phenomenon of masking, commonly
thought of as a mechanism by which the channels tuned to the signal are either "swamped"
(excitatory masking) or "suppressed" (suppressive masking) by the masker. Alternative
methods, quantifying masking as the degree to which a masker can disrupt the spatio-
temporal response pattern in response to a signal, might yield different results and/or
improve the accuracy in estimating frequency selectivity of methods currently used.
In summary, we investigated possible neural representation of the pitch of complex
tones and we proposed a spatio-temporal representation, which combines the advantages
and overcomes some of the limitations of strictly place and temporal codes, as the most
likely to be used in normal listening conditions. To our knowledge, this was the first study
in which a possible implementation of a spatio-temporal representation has been directly
tested physiologically.
Much work remains to be done to better understand the central
mechanisms of extraction of peripheral cues to pitch and the degree to which these
mechanisms might be similar across different species.
128
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